In 




1 



\ 



TREATISE 






S U ET" 





NG: 



IN "WHICH 



THE THEORY AND PRACTICE ARE FULLY 

EXPLAINED. 



PRECEDED BY 



A SHORT TREATISE ON LOGARITHMS: 



AND ALSO BY 



A COMPENDIOUS SYSTEM OF PLANE TRIGONOMETRY. 



®Ije bfyok lllnstraleb bg *§nmzxoms dkamples. 



BY 



SAMUEL ALSOP, 




PHILADELPHIA: 
E. C. & J. BIDDLE & CO., No. 508 MINOR ST. 

(Between Market and Chestnut, and Fifth and Sixth Sts.) 

1860. 



~Tt\5J\5 

. A46 
\8 6 



Entered according to act of Congress, in the year 1S57, by 

E. C. & J. DIDDLE, 

in the Clerk's Office of the District Court of the United States for the Eastern District of 

Pennsylvania. 



STEREOTYPED IiT L. J0HN8ON & CO. 
PHILADELPHIA. 



A Key to this -work has been published by E. C. & J. B. 



Let /g'7XSO 



PREFACE. 



The favor shown to this treatise by the author's colaborers in 
the educational field having called for another edition of it, he 
has carefully revised the work, and made such amendments as 
to him seemed desirable. These are not numerous, but he 
believes have somewhat improved the work. 

His aim has been to present the subject, in its practical as 
well as its theoretical relations, in a manner adapted to the capa- 
city of every student, by presenting the theory plainly and com- 
prehensively, and giving definite and precise directions for prac- 
tice ; and to embrace in the work every thing which an extensive 
business in land-surveying would be likely to require. How 
nearly his object has been attained, others must determine : he 
trusts, however, that the treatise will be found to possess merit 
sufficient to commend it to the favorable notice of his fellow- 
teachers. The following brief synopsis of its contents presents 
the plan and scope of the work. 

Chapter I. consists of a short explanation of the nature and 
use of Logarithms. 



6 PREFACE. 

Chapter II. contains the geometrical definitions and con- 
structions needed in the subsequent part of the work. 

In Chapter III. is presented a treatise on Plane Trigono- 
metry, including a great variety of examples illustrative of the 
solution of triangles. In this chapter will also be found a full 
description of the Theodolite and Surveyor's Transit, and direc- 
tions for their use. 

In Chapter IY. the principles of surveying by the Chain are 
explained. This method is little employed by practical sur- 
veyors in this country. Since, however, the measurements 
require no other instrument than a tape-line, or a cord, or some 
other means of determining distances, it is of importance to the 
farmer, who frequently desires to know- the contents of par- 
ticular fields, or of portions of enclosures. The second and 
third sections of this chapter contain a pretty full treatise on 
Field Geometry, or the method of performing on the ground, 
with the chain or measuring line only, those operations which 
are needed in fixing the positions of points or in locating lines. 
In Great Britain, Chain Surveying is almost exclusively em- 
ployed. 

Chapter Y. is devoted to Compass Surveying. Under this 
head are included all those methods which require the use of an 
instrument for determining the bearings of lines, whether that 
instrument be a Compass, a Transit, or a Theodolite. This 
chapter contains a full account of the methods to be employed 
in locating lines by means of such instruments. 

The numerous difficulties with which the surveyor will bo 
likely to meet from obstructions on the ground are stated, and 
the modes of overcoming them explained. 

This chapter, with that on Plane Trigonometry, constitutes, 
in fact, a full treatise on Surveying as practised in this country. 
In selecting the methods to be employed in overcoming the 
difficulties both in Compass and in Chain Surveying, care has 
been taken to adopt such only as maybe conveniently employed 
in the field. 

Chapter YI. contains the general principles of Triangular 



PREFACE. 7 

Surveying. This is the method employed in extensive geodetic 
operations. 

The details of this method are so complex that a volume — 
not a chapter — would be required for their development. All 
that has been attempted is to give some of the more simple 
principles. 

Chapter YII. treats of Laying out and Dividing Land. It is 
believed that many of the demonstrations in this chapter will be 
found to be much more simple than those usually given, almost 
all of them having been reduced to the development of a single 
principle. On a subject of this kind, which has so long occupied 
the attention of mathematicians, any thing new could hardly 
be expected. It has been the aim of the author to select the 
best methods, not to introduce any thing merely because it 
was new. 

Chapter IX. contains a treatise on Practical Astronomy, 
embracing all that is needed for the surveyor's purposes or is 
practicable with his instruments. Various methods of running 
meridian lines, and of determining the latitude and the time of 
day, are fully explained. 

The concluding chapter (X.) is devoted to the subject of 
the Variation of the Compass. In it will be found information 
of great value to the practical surveyor. The tables of varia- 
tion are in all cases drawn from the most recent and authentic 
sources. 

In the preparation of this treatise the author has consulted 
various well-known English and American mathematical works. 
To Professor Gillespie's excellent " Treatise on Land Surveying" 
(D. Appleton & Co., New York,) especially, the author is indebted 
for very valuable hints, particularly in the directions for prac- 
tice, the descriptions of the instruments, and various new 
methods of presenting important points. Some of these are 
referred to in their places. The typographical peculiarities of 
this volume, in the headings of articles, &c, were also suggested 
to the publishers by those of the work of Dr. Gillespie. 

In each department of the subject treated of in this volume 



8 PREFACE. 

the aim of the author has been to explain clearly the principles 
involved, and, as a general rule, to give only those methods for 
practice which he deems the best. By pursuing this course he 
has kept the volume within moderate limits, and has presented 
the subject in such a form as will, he trusts, meet the wants of 
teachers generally, as well as of very many practical surveyors. 
The tables appended to this treatise have been prepared with 
much care. That of Latitudes and Departures will be found to 
be more concise than those usually given, and, being extended 
to four decimal places, will enable the calculator to give greatei 
accuracy to his work. The table of Logarithms of Number? 
has been carefully compared with those of Babbage, Hutton, 
and other standard authors. That of Sines and Tangents was 
taken from Hutton, and compared with other seven-decimal 
tables. Besides, these, there is a table of Natural Sines and 
Cosines to every minute, and one of Chords to every five 
minutes, of the quadrant. 



CONTENTS. 



CHAPTER I. 

ON THE NATURE AND USE OF LOGARITHMS. 

Section 1. On the Nature of Logarithms. pagb. 

Definition and Illustration > 17 

Mode of calculating Logarithms 19 

Bases of Logarithms 19 

Indices of Logarithms..., 20 

Mantissse of Logarithms 20 

Description of the Table of Logarithms 20 

To find the Logarithm of a Number from the Table 21 

To find the Natural Number corresponding to a given Logarithm 23 

Section 2. On the Use of Logarithms. 

Multiplication by Logarithms 25 

Division by Logarithms 26 

Involution by Logarithms 27 

Evolution by Logarithms 27 

On the Use of Arithmetical Complements of Logarithms 28 

CHAPTER II. 

PRACTICAL GEOMETRY 

Section 1. Definitions 31 

Srotion 2. Geometrical Properties and Problems 36 

A. Geometrical Properties 36 

B. Geometrical Problems 39 

To bisect a given Straight Line 39 

To draw a Perpendicular to a Straight Line from a Point in it 40 

To let fall a Perpendicular to a Line from a Point without it 40 

At a given Point, to make an Angle equal to a given Angle 41 

To bisect a given Rectilineal Angle 42 

To draw a Straight Line touching a Circle 42 

Through a given Point to draw a Parallel to a given Straight Line 42 

To inscribe a Circle in a given Triangle 43 

To describe a Circle about a given Triangle 43 

To find a Third Proportional to two Straight Lines 43 

To find a Fourth Proportional to three Straight Lines 43 

To find a Mean Proportional between two Straight Lines 44 

To divide a Line into two Parts having a given Ratio 44 

9 



10 CONTENTS. 

CHAPTER III. 

PLANE TRIGONOMETRY. 

Section 1. Definitions. PAQE 

Measure of Angles. 45 

Trigonometrical Functions < 46 

Properties of Sines, Tangents, &c 47 

Geometrical Properties employed in Plane Trigonometry 48 

Section 2. Drafting or Platting. 

Mode of drawing Straight Lines 49 

Mode of drawing Parallels 49 

Mode of drawing Perpendiculars 51 

Mode of drawing Circles and Arcs 51 

Mode of laying off Angles with a Protractor 52 

By a Scale of Chords 52 

By a Table of Chords , 53 

Distances 53 

Drawing to a Scale 53 

Scales 55 

Diagonal Scale 55 

Proportional Scale 57 

Vernier Scale 57 

Section 3. Tables of Trigonometrical Functions. 

Description of the Table of Natural Sines and Cosines 58 

Description of the Table of Logarithmic Sines and Tangents 59 

Use of Table 60 

Table of Chords 63 

Section 4. On the Numerical Solution of Triangles. 

Definition 64 

The Numerical Solution of Right-Angled Triangles 64 

By the Use of the Table of Sines and Tangents 64 

By the Application of (47.1.) 66 

The Numerical Solution of Oblique- Angled Triangles. 

The Angles and one Side, or two Sides and an Angle opposite one 

of them, being given, to find the rest 67 

Two Sides and the included Angle being given, to find the rest. 

Rule 1 70 

Rule 2 71 

The three Sides being given, to find the Angles. 

Rule 1 73 

Rule 2 74 

Section 5. Instruments, and Field Operations. 

The Chain 76 

The Pins 78 

Chaining 78 

Recording the Outs 79 

Horizontal Measurement 80 

Tape Lines 82 

Angles 82 

The Transit and Theodolite. 

General Description 83 

The Telescope 87 



CONTENTS. 



11 



PAGE 

The Object Glass 88 

The Eve Piece 88 

The Spider Lines 89 

The Supports 91 

The Vertical Limb 91 

The Levels 92 

The Levelling Plates 92 

The Clamp and Tangent Screws 93 

The Watch Telescope 93 

Verniers « , 93 

The Reading of the Vernier 95 

To Read any Vernier 96 

Retrograde Verniers 96 

Reading backwards 98 

Double Verniers 98 

Adjustments 101 

First Adjustment: The Level should be parallel to the Horizontal Plates 102 
Second Adjustment : The Axis of the Horizontal Plates should be pa- 
rallel 102 

Third Adjustment: The Line of Collimation must be perpendicular 

to the Horizontal Axis \ 102 

The Line of Collimation in the Theodolite should be parallel to the 

Axis of the Cylinders on which the Telescope rests in its Ys 104 

Fourth Adjustment : The Horizontal Axis must be parallel to the 

Horizontal Plates 104 

Adjustments of the Vertical Limb 105 

First Adjustment : The Level must be parallel to the Line of Colli- 
mation 105 

Second Adjustment : The Zeros of the Vernier and Vertical Limb 

should coincide when the Telescope is horizontal 106 

Measuring Angles 107 

Repetition of Angles 108 

Verification of Angles 109 

Reduction to the Centre 109 

Angles of Elevation 110 

Section 6. Miscellaneous Problems to Illustrate the Rules of Plane Trigono- 
metry 110 

CHAPTEE IV. 



CHAIN SURVEYING. 

Section 1. Definitions. 

Definition 118 

Advantages 118 

Area Horizontal , , 119 

Section 2. Field Operations. 

Ranging out Lines 119 

To Interpolate Points in a Line 120 

On Level Ground 120 

Over a Hill 120 

By a Random Line 121 

Across a Valley 122 

To determine the Point of Intersection of two visual Lines 123 

To run a Line towards an invisible Intersection 123 

Perpendiculars. 

To draw a Perpendicular to a given Line from a Point in it. 

When the Point is accessible 123 



12 CONTENTS. 

PAGB 

When the Point is inaccessible 125 

To let fall a Perpendicular to a Line from a point without it. 

When the Point and Line are both accessible 125 

When the Point is remote or inaccessible 126 

When the Line is inaccessible 126 

The Surveyor's Cross 127 

To verify the Cross 128 

The Optical Square 128 

To test the Accuracy of the Square 129 

Parallels 

Through a given Point to draw a Parallel to an accessible Line 130 

To draw a Parallel to an inaccessible Line 130 

To draw a Parallel to a Line through an inaccessible Point 130 

Section 3. Obstacles in Running and Measuring Lines. 

To prolong a Line beyond an Obstacle 131 

To measure a line when both ends are accessible.... 132 

When one End is inaccessible 133 

When the inaccessible End is the intersection of two Lines 133 

When both Ends are inaccessible 134 

Section 4. Keeping Field Notes , 135 

Field Book 135 

Test Lines 139 

General Directions 139 

Platting the Survey 140 

Section 5. Surveying Fields of Particular Form. 

Rectangles 141 

Parallelograms 141 

Triangles. 

First Method 142 

Second Method 142 

Trapezoids 144 

Trapeziums. 

First Method 145 

Second Method 145 

Fields of more than four Sides. 

First Method 147 

Second Method 150 

Offsets 151 

Section 6. Tie Lines. 

Inaccessible Areas 159 

Defects of the Method 159 



CHAPTER V. 

■ COMPASS SURVEYING. 

Section 1. Definitions and Instruments. 

The Meridian 160 

The Points of the Compass 161 

Bearing 161 

Reverse Bearing 162 

The Magnetic Needle... 102 

The Magnetic Meridian • 163 



CONTENTS. 13 

PAGE 

The Magnetic Bearing 163 

The Compass 164 

The Sights ,, 166 

The Verniers 166 

The Pivot 168 

The Divided Circle 168 

Adjustments , ; 160 

Defects of the Compass 169 

Section 2. Field Operations. 

Bearings 170 

Use of the Vernier 171 

The Reverse Bearing 171 

Local Attraction 171 

To correct for Back Sights , 172 

By the Vernier 172 

To survey a Farm — General Directions. 172 

Random Line 173 

To determine the Bearing by a Station near.the Middle of the Line 174 

Proof Bearings 174 

Angles of Deflection ". 175 

Section 3. Obstacles in Compass Surveying. 

To run a Line making a given Angle with a given Line at a given 

Point within it 176 

To run a Line making a given Angle with a given inaccessible Line at a 

given Point in that Line . *. 177 

From a given Point out of a Line, to run a Line making a given Angle 
with that Line. 

If the Line be accessible 177 

If the Line be inaccessible 178 

If the Point be inaccessible, 178 

If the Point and the Line be both inaccessible. 179 

To run a Line parallel to a given Line through a given Point. 

If the Line and the Point be accessible 179 

If the Point be inaccessible 179 

If the Line be inaccessible 179 

If the Line and the Point both be inaccessible 180 

Prolongation and Interpolation of Lines 180 

To Prolong a Line beyond an Obstruction 181 

To Interpolate Points in a Line 182 

By a Random Line 182 

Measurement of Distances. 

To determine the Distance between two Points visible from each 

other 183 

To determine the Distance on a Line to the inaccessible but visible 

end » 185 

To determine the Distance when the end is invisible 186 

To determine the Distance to the Intersection of two Lines 186 

To determine the Distance between two inaccesi'ble Points 187 

Examples illustrative of the preceding Rules 188 

Section 4. Field Notes 190 

Section 5. Latitudes and Departures. 

Definitions 192 

The Bearing, Distance, Latitude, and Departure, — any two being given, 

to determine the others 193 

To determine the Latitude and Departure by the Traverse Table 194 

When the Bearing is given by Minutes 196 



14 CONTENTS. 

PAGE 

By the Table of Natural Sines and Cosines 197 

Test of the Accuracy of the Survey 199 

Correction of Latitudes and Departures 200 

Section 6. Platting the Survey. 

With the Protractor 202 

By a Scale of Chords 203 

By a Table of Natural Sines 204 

By a Table of Chords 205 

By Latitudes and Departures 205 

Section 7. Problems in Compass Surveying. 

Given the Bearing of one Side, and the Deflection of the next, to deter- 
mine its Bearing 208 

To determine the Deflection between two Courses 209 

To determine the Angle between two Lines 210 

To change the Bearings of the Sides of a Survey 211 

Section 8. Supplying Omissions. 

The Bearings and Distances bf all the Sides except one being given, to 

determine these .- 213 

All the Bearings and Distances except the Bearing of one Side and 

the Distance of another being given, to find these 217 

All the Bearings and Distances except two Distances being given, to find 

these 219 

All the Bearings and Distances except two Bearings being given, to find 

these 220 

Section 9. Content of Land. 

Given two Sides and the included Angle of a Triangle or Parallelogram, 

to find the Area 224 

The Angles and one Side of a Triangle being given, to find the Area 225 

To determine the Area of a Trapezium, three Sides and the two included 

Angles being given 226 

The Bearings and Distances of the Sides of a Tract of Land being 

given, to find its Area 229 

Offsets 235 

Inaccessible Areas 238 

Compass Surveying by Triangulation 243 



CHAPTER VI. 

TRIANGULAR surveying. 

Base 247 

Reduction to the Level of the Sea 248 

Signals 248 

Triangulation 248 

Base of Verification 250 

CHAPTER VIL 

LAYING OUT AND DIVIDING LAND. 

Section 1. Laying out land. 

To lay out a given Quantity of Land in the form of a Square 251 

To lay out a given Quantity of Land in the form of a Rectangle, one Side 

being given 251 

The Adjacent Sides having a given Ratio 252 



CONTENTS. 15 

PAGE 

One Side to exceed another by a given Difference 252 

To lay out a given quantity of Land in the form of a Triangle or Paral- 
lelogram, the Base being given 253 

One Side and the Adjacent Angle being given 253 

Lemma 254 

The Direction of two Adjacent Sides being given, to lay out a given 
quantity of land. 

By a Line running a given Course , 255 

By a Line running through a given Point 256 

Three Adjacent Sides of a Tract being given in Position, to lay out a 

given quantity of land 259 

By a Line parallel to the second Side 259 

By a Line running a given Course 262 

By a Line through a given Point 267 

By the shortest Line 269 

To cut off a Plat containing a given Area from a Tract of any number of 
Sides. 

By a Division line drawn from one of the Angles 269 

By a Line running a given Course 273 

To straighten Boundary lines 275 

To run a new Line between Tracts of different Values. 

By a Line running a given Course 280 

By a Line through a given Point in the old Line 281 

By a Line through a given Point in one of the Adjacent Sides 283 

Section 2. Division of Land. 

To divide a Triangle into two Parts having a given Ratio. 

By a Line through one of the Corners , 284 

By a Line through a Point in one of the Sides 284 

By a Line Parallel to one of the Sides 285 

By a Line running a given Course 286 

By a Line through a given Point 288 

To divide a Trapezoid into two parts having a given Ratio. 

By a Line cutting the Parallel Sides 290 

By a Line Parallel to the Parallel Sides 292 

To divide a Trapezium into two parts having a given Ratio. 

By a Line through a given Point on one Side 294 

By a Line through any Point 296 

By a Line Parallel to one Side , 298 

By a Line running a given Course 301 



CHAPTER VIII. 

MISCELLANEOUS EXAMPLES. 

Miscellaneous Examples 303 

CHAPTER IX. 

MERIDIANS, LATITUDE, AND TIME. 

Section 1. Meridians. 

Definition 307 

To run a Meridian Line. 

By equal Altitudes of the Sun 308 

By a Meridian Altitude of Polaris 309 

To determine the Time Polaris is on the Meridian 310 

To run a Meridian by a Meridian Passage observed with a Transit or 
Theodolite 314 



16 ' CONTENTS. 

PAGE 

By an Observation of Polaris at its greatest Elongation 314 

By Equal Altitudes of a Star 318 

Section 2. Latitude. 

To determine the Latitude by a Meridian Altitude of Polaris 319 

By a Meridian Altitude of the Sun 319 

By an Observation on a Star in the Prime Vertical 320 

Section 3. To find the Time of Day. 

By a Meridian Line 322 

By an observed Meridian Passage of a Star 322 

By an Altitude of the Sun or a Star not in the Meridian 323 

CHAPTER X. 

VARIATION OF THE COMPASS. 

Secular Change 325 

Table of Variations 326 

Line of no Variation 326 

To determine the Change in Variation by old Lines 327 

Diurnal Changes 329 

Irregular Changes 329 

APPENDIX. 

Demonstration of the Rule for finding the Area of a Triangle when three 
Sides are given 332 



TREATISE ON SURVEYING, 



CHAPTER I. 



ON THE NATURE AND USE OF LOGARITHMS. 



SECTION I. 

ON THE NATURE OF LOGARITHMS. 

1. Definition. Logarithms are a series of numbers, by 
the aid of which the operations of multiplication, division, 
the raising of powers, and the extraction of roots, may, 
respectively, be performed by addition, subtraction, multi- 
plication, and division. 

Such a series may be thus constructed. Above a geometric 
series, the first term of which is 1, place a corresponding 
arithmetic series, the first term of which is ; thus : — 

Arithmetical series, 0123456 7 8 
Geometrical series, 1 2 4 8 16 32 64 128 256 

To determine the product of any two terms of the geometric 
series, it is evidently only necessary to add the correspond- 
ing terms of the arithmetic series, and to notice the term of 
the geometric series agreeing to their sum ; which term is the 
product required. Thus, to find the product of 4 and 32, we 
add the corresponding terms, 2 and 5, in the arithmetic series. 
Their sum, 7, corresponds to 128, the product required. 

2. In a table of logarithms, the terms of the geometrical 
series are called the numbers; the ratio in this series is de- 
nominated the base of the table ; and the terms of the arith- 
metical series are called the logarithms of the corresponding 

2 17 



18 



THE NATURE AND USE OF LOGARITHMS. [Chap. I. 



terms of the geometric series. The numbers, it will be 
observed, are the powers of the base, and the logarithms are 
the indices of those powers. 

Further to illustrate the use of logarithms, we give the 
following table : — 



Num. 


Log. 


Num. 


Log. 


Num. 


Log. 
11 


2 


1 


64 


6 


2048 


4 


2 


128 


7 


4096 


12 


8 


3 


256 


8 


8192 


13 


16 


4 


512 


9 


16384 


14 


32 


5 


1024 


10 


32768 


15 



1. Eequired the quotient of 32768 divided by 2048. The 
indices or logarithms of these numbers are, respectively, 15 
and 11. The difference of these logarithms is 4, which is 
the logarithm of 16, the quotient required. Hence the 
difference of the logarithms of two numbers is the logarithm 
of their quotient. 

2. Required the third power of 32. The logarithm of 32 
is 5. Multiply this by 3, the index of the power to which 
32 is to be raised, and the product, 15, is the index of 32768, 
the required power. Hence, to involve a number to a given 
power, we multiply its logarithm by the index of the power 
to which it is to be raised. 

3. Required the fourth root of 4096. The index of this 
is 12. Divide this index by 4, the degree of the root to be 
extracted, and the quotient will be 3, which is the logarithm 
of 8, the root required. Hence, to extract the root of a 
number, we divide its logarithm by the number expressing 
the degree of the root to be extracted, and the quotient is 
the logarithm of the root required. 

3. The table in Art. 2 contains only the integral powers 
of 2, that being sufficient for the purpose of illustra- 
tion ; but a complete table contains all the numbers 
of the natural series, within the limits of the table, 
together with the indices, or logarithms. The logarithms 
in such a table will, in most instances, be fractions. 
Thus, the logarithms corresponding to any of the num- 
bers between 4 and 8 would be 2 and some fraction ; 



Sec. L] THE NATURE OF LOGARITHMS. 19 

of any number between 8 and 16, the logarithm would be 
3 and a fraction ; and so on. 

4. Calculation of Logarithms, Since all numbers are 
considered as the power $ some one base, we will have, 
if a be the base, and n tide *number, <2* = n. The deter- 
mination of the logarithm- will then'- consist in solving' the 
above equation so as to find x. This, in general, can only 
be done by approximation. The details to which it would 
lead are entirely foreign to the present work. Those who 
desire to become acquainted with the subject may consult 
the author's " Treatise on Algebra." 

5. Bases. Theoretically, it is of no importance what 
number is assumed as the base of the system ; but prac- 
tical convenience suggests that 10, the base of our system 
of notation, should also be the base of the system of loga- 
rithms. By the use of this base, it becomes unnecessary 
to insert in the table of logarithms their integral portions. 
For, as will be seen hereafter, the figures in the decimal por- 
tion of the logarithm depend on the figures in the number, 
while the integral portion of the logarithm depends solely 
on the position of the decimal point in the number. 

6. Assuming, then, 10 for a base, we have the following 
series : — 

lumbers, 1, 10, 100, 1000, 10000, 100000, 1000000; 
Logarithms, 12 3 4 5 6. 

The logarithm of any number between 1 and 10 will be 
wholly decimal; between 10 and 100, it will be 1 and a 
decimal ; and so on. 

If the powers of 10 be continued downwards, we have 

the powers 1 .1 .01 .001 .0001 .00001, 

and indices — 1 — 2 — 3 — 4 — 5. 

The logarithm of any number between .1 and 1 is there- 
fore — 1 + a decimal, of a number between .01 and .1 it is 
— 2 -j- a decimal, &c. 



20 THE NATURE AND USE OF LOGARITHMS. [Chap. I. 

7. Indices of Logarithms. The integral portion of 
every logarithm is called the index, the decimal portion 
being sometimes called the mantissa. From the above 
series, it is manifest that, if the number is greater than 1, 
the index is positive, and one less than the number of in- 
tegral figures. Thus, 246.75 coming between 100 and 
1000, its logarithm will be 2 and a decimal. If the num- 
ber is less than 1, the index will be negative. For ex- 
ample, the logarithm of .0024675, which comes between 
.001 and .01, will be — 3 + a decimal. 

8. Mantissae. The mantissas of logarithms to the base 
10 depend solely on the figures of the number, without 
any regard to the position of the decimal point. 

Let the logarithm of 31.416 be 1.497151 : then, since 
314.16 is 10 times 31.416, its logarithm will be 1.497151 + 
1 = 2.497151. Similarly, the logarithm of 31416, which is 
1000 times 31.416, will be 1.497151 + 3 = 4.497151. 

Again, .031416 = 31.416 -f- 1000 : its logarithm is there- 
fore 1.497151 — 3 = —2.497151, in which the sign — is 
understood to belong solely to the index 2, and not to the 
mantissa. Since, then, the index can be supplied by atten- 
tion to the position of the decimal point, the mantissae 
alone are inserted in the body of a table of logarithms. 

The annexed table will illustrate the above more fully : — 



Number. 


Logarithm. 


64790 


4.811508 


6479 


3.811508 


647.9 


2.811508 


64.79 


1.811508 


6.479 


0.811508 


.6479 


—1.811508 


.06479 


—2.811508 


.006479 


—3.811508 



9. Table of Logarithms. A table of logarithms consists 
of the series of natural numbers, with their logarithms, or, 
rather, the mantissae of their logarithms, so arranged that 



Sec. I.] 



THE NATURE OF LOGARITHMS. 



21 



one can be readily determined from the other. In the 
table of logarithms appended to this treatise, the mantissas 
of the logarithms of all numbers, from 1 to 9999 inclusive, 
are given. On the first page are found the numbers from 
1 to 99, with their logarithms in full. The remaining pages 
contain only the mantissas of the logarithms. The first 
column, headed N", contains the numbers, from 100 to 999 ; 
and the second, headed 0, the mantissas of their logarithms. 
Thus, the logarithm of the number 897 is 2.952792 ; the 
index being 2, because there are three integral figures in 
the number. 

The remaining columns contain the last four figures of 
the mantissas of the logarithms of numbers of four figures, 
the first three of which are found in the first column, and 
the fourth, at the head. Thus, if the number were 8976, 
the last four figures 3083 of the mantissa of its loga- 
rithm would be found in the column headed 6 ; the first 
two, 95, found in the second column, being common to 
them all. The logarithm of 8976 is, therefore, 3.953083. 

10. To denote the point in which the second figure 
changes, when such change does not take place in the first 
logarithmic column, the first of the four figures from the 
change to the end of the line is printed as an index figure ; 
thus, on page 25 of the tables, we have the lines 



N. 





l 


2 


3 


4 


5 


6 


7 


8 


9 


456 
457 

458 


8965 

9916 

660865 


9060 
°011 
0960 


9155 
°106 
1055 


9250 
°201 
1150 


9346 
°296 
1245 


9441 
°391 
1339 


9536 
°486 
1434 


9631 
°581 
1529 


9726 
°676 
1623 


9821 
°771 
1718 



In such cases the first two figures are found in the next 
line. The logarithm of 4575 is, therefore, 3.660391. 



11. To find the Logarithm of a number from the 
tables. If the number consists of one or two figures only, 
its logarithm is found on the first page of the table. If the 
two figures are both integers, the index is given also ; but, 
if the one or both figures be decimal, the decimal part only 



22 THE NATURE AND USE OF LOGARITHMS. [Chap. I. 

of the logarithm should be taken out. Thus, the loga- 
rithm of 8 is 0.903090; of 59 is 1.770852. 

If the number be wholly or part a decimal, the index 
must be changed in accordance with the principles laid 
down in Art. 7. Thus, the index must be one less than the 
number of figures in the integral part of the natural num- 
ber. But when the natural number is wholly a decimal 
the index is negative, and must be one more than the num- 
ber of ciphers between the first significant figure and the 
decimal point. Thus, the logarithm of 

.8 is —1.903090 ; of .059 is —2.770852. 

If the number consists of three figures, look for it in the 
remaining pages of the table, in the column headed N". 
Opposite to it, in the first column, will be found the deci- 
mal portion of the logarithm ; the first two figures of the 
logarithm, being common to all the columns, are printed 
but once, to save room. Thus, the logarithm of 

272 is 2.434569 ; of 529 is 2.723456 ; 

the index being placed in accordance with the above rule. 

If the number consists of four figures, the first three 
must be found as before ; and the fourth, at the top of the 
table. The last four figures of the logarithm are found 
opposite to the first three figures of the number, and under 
the fourth ; the first two figures of the logarithm being 
found in the first logarithmic column. Thus, if the num- 
ber were 445.8, look for 445 in the column headed !N", and 
opposite thereto, in the column headed 8, the figures 9140 
are found; these affixed to 64, found in the first column, 
give 649140 for the decimal portion of the logarithm ; and, 
as there are three integral figures, the index is 2. Hence, 
the complete logarithm is 2.649140. 

If there are more than four figures in the number, find 
the logarithm of the first four figures as before. Take the 
difference between this logarithm and the next greater in 
the table ; multiply this difference by the remaining figures 
in the number, and from the product separate as many 
figures from the right hand as are contained in the mul- 



Sec. L] THE NATURE OF LOGARITHMS. 23 

tiplier ; then add the remainder to the logarithm first taken 
out : the sum will be the required logarithm. 
Let the logarithm of 6475.48 be required. 

The logarithm of 6475 is .811240 
The next greater is 1307 

' 67 
67 x 48 = 32,16 

32 added to 811240 gives .811272 ; 

and the index being 3, the complete logarithm is 3.811272. 

Next let the logarithm of .0026579 be required. 

The logarithm of 2657 is .424392 
The next greater 4555 

Difference 163 

9 



146,7 
424392 + 147 = .424539, and the index being -3. the com- 
plete logarithm is —3.424539. 

Note. — In this last example, the product is 1467 : the figure stricken off 
being 7, which is more than 5, 147 is taken instead of 146. 

Examples. 
Required the logarithms of the following numbers : — 

1. Of 7.5 0.875061 

2. Of 876 2.942504 



3. Of 93.37 1.970207 

4. Of .4725 —1.674402 

5. Of .869427 —1.939233 

6. Of .01367 —2.135769 



7. Of .0645775 —2.810081 

8. Of .004679 —3.670153 

9. Of 37196.2 4.570499 

10. Of .14638 —1.165482 

11. Of 6273.69 3.797523 

12. Of .037429 —2.573208 



12. To find the natural number corresponding to a 
given Logarithm. If four figures only be needed in the 
answer, seek in the columns of logarithms for the one near- 
est to the decimal part of the given logarithm : the first 
three figures of the natural number will be found in the 
column marked N ; and the fourth, at the top of the column 
in which the logarithm is found. 

When the index is positive, the number of integral 



24 THE NATURE AND USE OF LOGARITHMS. [Chap. I. 

figures will be one greater than the number expressed by 
the index ; but, if the index is negative, the number will 
be wholly decimal, and have one less cipher between the 
decimal point and the first significant figure than the num- 
ber expressed by the index. Thus, the natural number 
corresponding to the logarithm 2.860996 is 726.1; and that 
corresponding to —2.860996 is .07261. 

If the logarithm be found exactly in the tables, and there 
be not enough figures in the corresponding number, the 
deficiency must be supplied by ciphers. Thus, the natural 
number corresponding to 6.891649 is 7792000. 

But, if fi.ve or six figures be required, find in the table 
the logarithm next less than the given one, and take out 
the corresponding number as before ; subtract this loga- 
rithm from the next greater in the table, and also from the 
given logarithm; annex one or two ciphers to the latter 
remainder, according as ^.ve or six figures are required, and 
divide the result by the former. The quotient annexed to 
the figures first taken out will give the figures required, 
the decimal point being placed as before. 

Required the number corresponding to 2.649378, to six 
figures 



Given logarithm 

Next less 


.649378 

.649335 cor. num. 4460 


Difference 


43 


Next greater logarithm 
Next less 


.649432 
.649335 


Difference 


97)4300(44 

388 




420 




388 



Hence, the number is 446.044. 



32 



Examples. 

Required the natural numbers corresponding to the fol- 
lowing logarithms. 



Sec. II.] 



ON THE USE OF LOGARITHMS. 



25 



1. 2.467415 

2. —1.396143 

3. 2.04163T 

4. —3.167149 



Ans. 293.37 
.24897 
110.062 
.0014694 



5. 4.617392 

6. 1.947138 

7. —2.960014 

8. —2.760116 



Ans. 41437.3 
88.54 
.091204 
.057559 



SECTION II. 

ON THE USE OF LOGARITHMS. 

13. Multiplication. To multiply numbers by means of 
logarithms. Add together the logarithms of the factors, 
and take out the natural number corresponding to the 
sum. If any of the indices be negative, the figure to be 
carried from the sum of the decimal portions must be con- 
sidered positive, and added to the sum of the positive, or 
subtracted from the sum of the negative indices. Then 
collect the aflirmative indices into one sum, and the nega- 
tive into another, take the difference between these sums, 
and prefix thereto the sign of the greater sum. 

Examples. 
Ex. 1. Multiply 47.25 and 397.3. 





47.25 


log. 1.674402 




397.3 


" 2.599119 


Product, 


18772.5 


4.273521 


Ex. 2. 


Required the 


product of 764.3, .8175, .04729, and 


.00125. 








764.3 


log. 2.883264 




.8175 


" —1.912488 




.04729 


« —2.674769 




.00125 


« —3.096910 


Product, 


.0369344 


—2.567431 



Ex. 3. Required the product of 87.5 and 6.7. 

Ans. 586.25. 



26 



THE NATURE AND USE OF LOGARITHMS. [Chap. I. 



Ex. 4. Required the continued product of .0625, 41.67, 
.81427, and 2.1463. Ans. 4.5516. 

Ex. 5. Multiply 67.594, .8739, and 463.92 together. 

Ans. 27404. 

Ex. 6. Multiply 46.75, .841, .037654, and .5273 together. 

Ans. .780633. 
Ex. 7. Multiply .00314, 16.2587, .32734, .05642, and 1.7638 
together. 



Ans. .001663. 



14. Division. To divide numbers by logarithms. Subtract 
the logarithm of the divisor from that of the dividend : the 
remainder will be the logarithm of the quotient. 

If one or both of the indices are negative, subtract the 
decimal portions of the logarithm as before ; and, if there 
be one to carry from the last figure, add it to the index of 
the divisor, if this be positive, but subtract if it be nega- 
tive ; then conceive the sign of the result to be changed, 
and if, when so changed, the two indices have the same 
sign, add them together ; but, if they have different signs, 
take their difference and prefix the sign of the greater. 





Examples. 




Ex. 1. Divide 


6740 


log. 


3.828660 


fey 


87 


log. 


1.939519 


Quotient, 77.471 






1.889141 


Ex. 2. Divide 


86.47 


log. 


1.936865 


fey 


.0124 


log. - 


-2.093422 


Quotient, 6973.4 






3.843443 


Ex. 3. Divide 


.0642 


log. - 


-2.807535 


fey 


87.63 


log. 


1.942653 


Quotient, .00073263 




-4.864882 


Ex. 4. Divide 


.0642 


log. - 


-2.807535 


fey 


.008763 


log. - 


-3.942653 


Quotient, 7.3263 






0.864882 


Ex. 5. Divide 407.3 by 27.564. 


Ans. 14.7765 


Ex. 6. Divide . 


80743 by 


63.87. 


Ans. .012642 



Sec. II. ] ON THE USE OF LOGARITHMS. 27 

Ex. 7. Divide 963.T by .00416. Ans. 231659. 

Ex. 8. Divide 86.39 by .09427. Ans. 916.41. 

Ex. 9. Divide .006357 by .0574. Ans. .11075. 

Ex. 10. Divide 76.342 by .09427. Ans. 809.82. 

15. To involve a number to a power. Multiply the 
logarithm of the number by the index of the power to 
which it is to be raised. 

If the index of the logarithm is negative, and there is 
any thing to be carried from the product of the decimal 
part by the multiplier, instead of adding this to the pro- 
duct of the index, subtract it: the difference will be the 
index of the product, and will always be negative. 

Ex. 1. Eequired the fourth power of 5.5. 

5.5 log. 0.740363 

4 

915.065 2.961452. 

Ex. 2. Eequired the fifth power of .63. 

.63 log. —1.799341 

5 

.099244 —2.996705. 

Ex. 3. Eequired the fourth power of 7.639. 

Ans. 3405.24. 

Ex. 4. Eequired the third power of .03275. 

Ans. .00003513. 

Ex. 5. What is the fifteenth power of 1.06 ? 

Ans. 2.3966. 

Ex. 6. What is the sixth power of .1362 ? 

Ans. .0000063836. 

Ex. 7. What is the tenth power of .9637? 

Ans. .69091. 

16. To extract a given root of a number. Divide the 
logarithm of the number by the degree of the root to be 
extracted : the quotient will be the logarithm of the root. 

If the index of the logarithm is negative, and does not 



28 THE NATURE AND USE OF LOGARITHMS. [Chap. I. 

contain the divisor an exact number of times, increase it 
by so many as are necessary to make it do so, and carry 
the number so borrowed, as so many tens to the first figure 
of the decimal. 

Ex. 1. Extract the fourth root of 56.372. 
56.372 log. 4) 1.751063 

Result, 2.7401 .437766 

Ex. 2. Extract the fifth root of .000763. 

.000763 log. 5) — 4.882525 

Result, .23796 —1.376505. 

Ex. 3. What is the fifth root of .00417 ? Ans. .3342. 

Ex. 4. Required the fourth root of .419. Ans. .80455. 
Ex. 5. Required the tenth root of 8764.5. Ans. 2.479. 
Ex. 6. Required the seventh root of .046375. 

Ans. .6449. 

Ex. 7. Required the fifth root of .84392. Ans. .96663. 
Ex. 8. Required the sixth root of .0043667. Ans. .40429. 

17. Arithmetical Complements. When several num- 
bers are to be added, and others subtracted from the sum, 
it is often more convenient to perform the operation as 
though it were a simple case of addition. This may be 
done by conceiving each subtractive quantity to be taken 
from a unit of the next higher order than any to be found 
among the numbers employed ; then add the results with 
the additive numbers, and deduct from the result as many 
units of the order mentioned as there were subtractive 
numbers. The difference between any number and a unit 
of the next higher order than the highest it contains is 
called the arithmetical complement of the number. Thus, the 
arithmetical complement of 8765 is 1235. It is easily ob- 
tained by taking the first significant figure on the right from 
ten, and each of the others from nine. This may be done 
mentally, so that the arithmetical complements need not be 
written down. 

Thus, suppose A started out with 375 dollars to collect 



Sec. II. ] 



ON THE USE OF LOGARITHMS. 



29 



some bills and to pay sundry debts. From B he received 
$104, to D be pays $215, to E be pays $75, from F be re- 
ceives $437, and, finally, pays to G $137. How mucb has 
he left? 



375^ 
104 
—215 

— 75 

437 

—137 

Ans. 489 



r 



which are added as 
though they were 



f375 
104 

785 
925 
437 

863 

v. | 

3489, 



deducting 3000 from the final result 3489, because there 
were three subtractive quantities. 

The arithmetical complements of logarithms are gene- 
rally employed where there are more subtractive logarithms 
than one. To give symmetry to the result, it would be 
neater to employ them in all cases. To a person who has 
much facility in calculation, it is most convenient to write 
down the logarithm as taken from the table, and obtain 
the arithmetical complement as the work is carried on. 
Thus, in the example above, the numbers could be written 
as in the first column ; but in the addition, instead of em- 
ploying the figures as they appear in the subtractive num- 
ber, the complement of the first significant figure to ten, 
and of the others to nine, should be employed. 

As an example of the use of the arithmetical comple- 
ments of the logarithms of numbers, let it be required to 



27 475 
work by logarithms the proportion as — : -j=- 



125 : x. 



Here, as the first term is a fraction, it will have to be in- 
verted; and the question will be the same as finding the 
„ 5o x 475 x 125 



u ^ *-'- L 


27 x 17 










log. 27 


/ 1.431364") 

\ 1.230449 


which are 


fA. 


C. 


8.568636 


" 17 


added as 


A. 


c. 


8.769551 


" 55 


1.740363 


\- though 


[ 




1.740363 


" 475 


2.676694 


they were 






2.676694 


" 125 


2.096910 J 


written 


< 




2.096910 


Result, 7114.66 3.852154 








3.852154 



30 THE NATURE AND USE OF LOGARITHMS. [Chap. I. 

deducting 20, because there were two arithmetical comple- 
ments employed. 

In the examples wrought out in the subsequent part of 
this work, the arithmetical complements of the logarithms 
of the first term of every proportion are employed. 



CHAPTER II. 



PRACTICAL GEOMETRY. 



SECTION I. 

DEFINITIONS. 

18. The practical surveyor will find a good knowledge 
of Algebra and of the Elements of Geometry an invaluable 
aid not only in elucidating the principles of the science, 
but in enabling him to overcome difficulties with which he 
will be certain to meet. In fact, so completely is Survey- 
ing dependent on geometrical principles, that no one can 
obtain other than a mere practical knowledge of it, without 
first having mastered them ; and he who depends solely 
on his practical experience will be certain to meet with 
cases which will call for a kind of knowledge which he 
does not possess, and which he can obtain only from 
Geometry. 

Every student, therefore, who desires to become an in- 
telligent surveyor, should first study Euclid, or some other 
treatise on Geometr}^. He will then have a key which will 
not only unlock the mysteries contained in the ordinary 
practice, but which will also open the way to the solution 
of all the more difficult cases which occur. To those who 
have taken the course above recommended, the problems 
solved in the present chapter will be familiar. They are 
inserted for the benefit of those who may not be thus pre- 
pared, and also as affording some of the most convenient 
modes of performing the operations on the ground. 



19. Geometry is the science of magnitude and position. 

31 



32 PRACTICAL GEOMETRY. [Chap. II. 

20. A solid is a magnitude having length, breadth, and 
thickness. 

All material bodies are solids, and so are all portions of 
space, whether they are occupied with material substances 
or not. Geometry, treating only of dimension and posi- 
tion, has no reference to the physical properties of matter. 

21. The surfaces of solids are superficies. A superficies 
has, therefore, only length and breadth. 

22. The boundaries of superficies, and the intersection 
of superficies, are lines. Hence, a line has length only. 

23. The extremities of lines, and the intersections of 
lines, are points. A point has, therefore, neither length, 
breadth, or thickness. 

24. A point, therefore, may be defined as that which has 
position, but not magnitude. 

25. A line is that which has length only. 

26. A straight line is one the direction of which does not 
change. It is the shortest line that can be drawn between 
two points. 

27. A superficies has length and breadth only. 

28. A plane superficies, generally called simply a plane, is 
one with which a straight line may be made to coincide in 
any direction. 

29. A plane rectilineal angle, or sim- 
ply an angle, is the inclination of 
two lines which meet each other. 
(Fig- I-) 

A. 

30. An angle may be read either by the single letter at 




Sec. I.] 



DEFINITIONS. 



33 




the intersection of the lines, or by three letters, of which 
that at the intersection must always occupy the middle. 
Thus, (Fig. 1,) the angle between BA and AC may be read 
simply A or BAC. 

31. The magnitude of an angle has no reference to the 
space included between the lines, nor to their length, but 
solely to their inclination. 

32. Where one straight line stands on another so as to 
make the adjacent angles equal, Fig# 2 . 

each of these angles is called a 
right angle; and the lines are said 
to be perpendicular to each other. 
Thus, (Fig. 2,) if ACD = BCD, 
each is a right angle, and CD is 
perpendicular to AB. a c b 

33. An angle less than a right angle is called an acute 
angle. Thus, BCE or ECD (Fig. 2) is an acute angle. 

34. An angle greater than a right angle is called an 
obtuse angle. ACE (Fig. 2) is an obtuse angle. 

35. The distance of a point from a straight line is the 
length of the perpendicular from that point to the line. 

36. Parallel straight lines are those of which all points 
in the one are equidistant from the other. 

37. A figure is an enclosed space. 

38. A triangle is a figure bounded by three straight lines. 

39. An equilateral triangle is one the three sides of which 
are equal. 

40. An isosceles triangle is one of which two of the sides 
are equal. The third side is called the base. 

3 



34 PRACTICAL GEOMETRY. [Chap. II. 

41. A scalene triangle has three unequal sides. 

42. A right-angled triangle has one of its angles a right 
angle. 

43. The side opposite the right angle is called the hypo- 
thenuse, and the other sides, the legs. 

44. An obtuse-angled triangle has one of its angles obtuse. 

45. A quadrilateral figure is bounded by four sides. 



Fig. 3. 



46. A parallelogram (Fig. 3) is a 
quadrilateral, the opposite sides of 
which are parallel. 




47. A rectangle (Fig. 4) is a parallelogram, the adjacent 
sides of which are perpendicular to each Fig. 4. 
other. Thus, ABCD is a rectangle. A 
rectangle is read either by naming the 
letters around it in their order, or by 
naming two of the sides adjacent to any 
angle. Thus, the rectangle ABCD is ^ 
read the rectangle AB.BC. 

Whenever the rectangle of two lines, such as DE.EF, is 
spoken of, a rectangular parallelogram, the adjacent sides 
of which are equal to the lines DE and EF, is meant. 

48. A square is a rectangle, all the sides of which are 
equal. 

49. A rhombus is an oblique parallelogram, the sides of 
which are equal. 

50. A rhomboid is an oblique parallelogram, the adjacent 
sides of which are unequal. 



Sec. L] 



DEFINITIONS. 



35 



51. All quadrilaterals that are not parallelograms are 
called trapeziums. 

52. A trapezoid is a trapezium, having two of its sides 
parallel. 

53. Figures of any number of sides are called polygons, 
though this term is generally restricted to those having 
more than four sides. 

54. The diagonal of a figure is a line joining any two 
opposite angles. 



Fig. 5., 



55. The base of any figure is the 
side on which it may be supposed 
to stand. Thus, AB (Fig. 5) is the 
base of ABCD. 




56. The altitude of a figure is the distance of the highest 
point from the line of the base. CE (Fig. 5) is the altitude 
of ABCD. 

57. The diameter of a circle is a straight line through the 
centre, terminating in the circumference. 

58. The radius of a circle is a straight line drawn from 
the centre to the circumference. 



Fig. 6. 



59. A segment of a circle is any part 
cut off by a straight line. Thus, 
ABCD is a segment. 




36 



PRACTICAL GEOMETRY. 



[Chap. II. 



60. A semicircle is a segment cut oft* 
by the diameter. ABC and AEB (Fig. 
7) are semicircles. 




61. A quadrant is a portion of a circle included between 
two radii at right angles to each other. ADCandBDC 
(Fig. 7) are quadrants. 

62. The angle in a segment is the angle contained between 
two straight lines drawn from any point in the arc of a seg- 
ment to the extremities of that arc. Thus, ABD and ACD 
(Fig. 6) are angles in the segment ABCD. 

63. Similar rectilineal figures have their angles equal, 
and the sides about the equal angles proportionals. 

64. Similar segments of a circle are those which contain 
equal angles. 



SECTION II. 

GEOMETRICAL PROPERTIES AND PROBLEMS. 

A— GEOMETRICAL PROPERTIES. 

65. All right angles are equal to each other. 

66. The angles which one straight line makes with an- 
other on one side of it are together equal to two right 
angles. Thus, ACE and ECB (Fig. 2) are together equal to 
two right angles. (13.1.) 



Sec. II.] GEOMETRICAL PROPERTIES AND PROBLEMS. 



67- If a number of straight lines are drawn from a point 
in another straight line, all the successive angles are together 
equal to two right angles. Thus, A CD + DCE + ECB (Fig. 
2) make two right angles. 



68. If two straight lines inter- 
sect each other, the angles verti- 
cally opposite are equal. Thus, 
AEC (Fig. 8) = BED, and AED = 
BEC. (15.1.) 



Fig. 8. 




69. Triangles which have two sides and the included 
angle of one respectively equal to the two sides and the 
included angle of the other, are equal in all respects. (4.1.) 

70. Triangles which have two angles and the interjacent 
side of one respectively equal to two angles and the inter- 
jacent side of the other, are equal in all respects. (26.1.) 

71. Triangles which, have two angles of the one respec- 
tively equal to two angles of the other, and which have also 
the sides opposite to two equal angles equal to each other, 
are equal in all respects. (26.1.) 



72. If a straight line cuts two pa- 
rallel lines, the angles similarly situ- 
ated in respect to these lines, and 
also those alternately situated, will be 
equal to each other (29.1.) Thus, 
(Fig. 9,) EFB = FGD, BFG = DGH, 
AFE = CGF, and AFG = CGH, 
being similarly situated ; and AFE 
= DGH, EFB = CGH, AFG = 
FGD, and BFG = FGC, being alternately situated. 




B 



73. If a straight line cuts two parallel straight lines, the 
two exterior angles on the same side of the cutting line, 
and also the two interior angles, are equal to two right 




38 PRACTICAL GEOMETRY. [Chap. II. 

angles. Thus, (Fig. 9,) EFB and DGII are equal to two 
right angles, as are also AFE and CGH. So also the pairs 
of interior angles AFG and FGC, BFG and FGD, are each 
equal to two right angles. (29.1.) 

74. The angles at the base of an isosceles triangle are 
equal to each other. (5.1.) 

75. If one side of a triangle be 
produced, the exterior angle so 
formed will be equal to .the two 
angles adjacent to the opposite side, 
and the three interior angles are 
equal to two right angles. Thus, 
(Fig. 10,) ACD = ABC + BAC, and 
ABC + BAC + ACB = two right angles. (32.1.) 

76. The interior angles of any rectilineal figure are equal 
to twice as many right angles as the figure has sides, dimi- 
nished by four right angles. The interior angles of a quadri- 
lateral are therefore equal to four right angles. (Cor. 1, 
32.1.) 

77. The opposite sides and angles of a parallelogram are 
equal to each other. (34.1.) 

78. Conversely, any quadrilateral of which the opposite 
sides or the opposite angles are equal is a parallelogram. 

79. Parallelograms having equal bases and altitudes, and 
also triangles having equal bases and altitudes, are equal to 
each other. (35-38.1.) 

80. A parallelogram is double a triangle having the same 
base and altitude. (41.1.) 

81. The square on the hypothenuse of a right-angled 
triangle is equal to the sum of the squares of the legs. 
(47.1.) 



Sec. II.] GEOMETRICAL PROPERTIES AND PROBLEMS. 



39 



82. Any figure described on the hypothenuse of a right- 
angled triangle is equal to the sum of the similar figures 
similarly described on the sides. (31.6.) 



Fig. 11. 



83. The angle at the centre of a 
circle is double the angle at the cir- 
cumference on the same base. Thus, 
the angle at C (Fig. 11) is double 
either D or E. (20.3.) 




84. Angles in the same segment of a circle are equal. 
Thus, D and E (Fig. 11) are equal. 

85. The angle in a semicircle is a right angle ; the angle 
in a segment greater than a semicircle is acute ; and that in 
a segment less than a semicircle is obtuse. 

86. The sides about the equal angles of equiangular tri- 
angles are proportional. (4.6.) 



B .— GEOMETRICAL PROBLEMS. 

Under this head are given those methods of construction 
which are applicable to paper drawings. The methods to 
be used in field operations will be given in a subsequent 
chapter. 



87. Problem 1. — To bisect a given 
straight line. Let AB (Fig. 12) be the 
given line. With the centres A and 
B, and radius greater than half AB, 
describe arcs cutting in C and D. 
Join CD cutting AB in E, and the 
thing is done. (10.1.) 



Fig. 12. 






s^ 



40 



PRACTICAL GEOMETRY. 



[Chap. II. 



Problem 2. To draw a perpendicular to a straight line from 
a given point in it. 
a. When the point is not near the end. 



88. Let AB (Fig. 13) be the line and 
the given point. Lay off CD = CE, and 
with D and E as centres, and any radius 
greater than DC, describe arcs cutting in 
F. Draw CF, and the thing is done. 

(ii.i) 



Fig. 13. 



,-¥' F 



b. When the point is near the end of the line. 

89. First Method. — Take any point 
D (Fig. 14) not in the line, and with 
the centre D and radius DC de- 
scribe the circle ECF, cutting AB in 
E. Join ED and produce it to F. 
Then will CF be the perpendicular. 
For ECF, being an angle in a semi- ~~ 
circle, is a right angle. (85.) 



90. Second Method.— With C 
(Fig. 15) and any radius describe 
DEF ; with D and the same radius 
cross the circle in E ; and with E 
as a centre, and the same radius, 
cross it in F. With E and F as 
centres, and any radius, describe 
arcs cutting in G. Then will CG 
be the perpendicular. 



Fig. 14. 



d/ 



E N \ ^ 



>'o, 



Fig. 15. 



>Cg 



E v .^ — 






C B 



Problem 3. — To let fall a perpendicular to a line from a point 
without it. 
a. When the point is not nearly opposite the end of the line. 



Sec. II. ] GEOMETRICAL PROPERTIES AND PROBLEMS. 



41 



91. Let AB (Fig. 16) be the line 
and C the given point. With the 
centre C describe an arc cutting AB 
in D and E. With the centres D and 
E and any radius describe arcs cut- 
ting in F. Join CF, and the thing 
is done. (12.1.) 



A 



Fig. 16. 
C 



\ 


G 


D\ 


X E 


V 


<* 



B 



b. When the point is nearly opposite the end of the line. 

Fig. 17. 

92. First Method.— With D and E 
as centres, and radii DC and EC, de- 
scribe arcs cutting in F : then will CF 
be the perpendicular. For, the tri- 
angles CDE and FDE being equal, 
(8.1,) DGC and FGD will be equal. 
(4.1.) 



A- 



C 



E\ 



-f— B 



^ 



93. Second Method.— Lei F (Fig. 14) be the point. From 
F to any point E in the line AB draw FE. On it describe 
a semicircle cutting AB in C. Join F and C, and FC will 
be the perpendicular (85.) 

Problem 4. — At a given point in a given straight line to 
make an angle equal to a given angle. 

94. Let BCD (Fig. 18) be the given 
angle, and A the given point in AE. 
With the centre C and any radius de- 
scribe BD, cutting the sides of the angle 
in B and D. With A as a centre and 
the same radius describe EF ; make EF 
= DB ; draw AF, and the thing is done. 




42 



PRACTICAL GEOMETRY. 



[Chap. II. 




Problem 5. — To bisect a given angle. 

95. Let BAC (Fig. 19) be the given 
angle. With the centre A and any radius 
describe an arc cutting the sides in B and C. 
With the centres B and C, and the same or 
any other radius, describe arcs cutting in 
D. Join AD, and the thing is done. (9.1.) 



Problem 6. — To draw a straight line touching a circle from 
a given point without it. 

96. Let ABC be the given 
circle, and D the given point. 
Join D and the centre E. On 
DE describe a semicircle cut- 
ting the circumference in B. 
Join DB, and it will be the tan- 
gent required. 

For DBE, being an angle in a semicircle, is a right angle, 
(31.3 ;) therefore, DB touches the circle, (16.3.) 

If the point were in the circumference at B. Join EB, 
and draw BD perpendicular to it. BD will be the tangent. 

Problem 7. — Through a given point to draw a line parallel 
to a given straight line. 

97. First Method.— Let A (Fig. 21) 
be the given point, and BC the given 
line. From A to BC let fall a per- 
pendicular AD ; and at any other 
point E in BC erect a perpendicular 
EF equal to AD. Through A and F draw AF, which will 
be the parallel required. 




Fig. 21. 



B E 



D 



98. Second Method. — From A (Fig. 
22) to D, any point in BC, draw AD. 
Make DAE = ADC, and AE will be 
parallel to BC. (2T.1.) 



Fig. 22. 




Sec. II.] GEOMETRICAL PROPERTIES AND PROBLEMS. 



43 



99. Third Method. — Through A draw 
ADE, cutting BC in D. Make DE = 
AD. Through E draw any other line 
EFG, cutting BC in F. Make FG = 
EF : then AG will be parallel to BC. 
(2.6.) 




Problem 8. — To inscribe a circle in a given triangle. 



100. Let ABC (Fig. 24) be the 
given triangle. Bisect two of its 
angles A and B by the lines AD, 
BD, cutting in D. Then will D be 
the centre. (4.4.) 



Problem 9. — To describe a circle about a given triangle. 

Fig. 25. 

101. Bisect two of the sides, as AC 
and AB, (Fig. 25,) by the perpendicu- 
lars FE and DE, cutting in E. Then 
will E be the centre of the required 
circle. 





Fig. 26. 



Problem 10. — To find a third proportional to two straight 
lines. 

102. Let M and S" (Fig. 26) be 
the given lines. Draw two lines 
AB and AC, making any angle at 
A. Lay off AD — M, and AE and 
AF each equal to N. Join DF, 
and draw EG parallel to it. AG 
will be the third proportional re- 
quired. (11.6.) 




E D 



M- 
N- 



Problem 11. — To find a fourth proportional to three given 

straight lines. 



44 



PRACTICAL GEOMETRY. 



[Chap. IL 



103. Let M, N, and (Fig. 27) 
be the three lines. Draw any two 
lines AB and AC, meeting at A. 
Lay off AD = M, AE = N, and AF 
= 0. Join DF, and draw EG pa- 
rallel to it : then AG is the fourth 
proportional required. (12.6.) 



Fig. 27. 




M 

N 




Fig. 28. 




Problem 12. — To find a mean proportional between two 
straight lines. 

104. First Method. — Place the lines 
AB and BC (Fig. 28) in the same 
straight line. On AC describe a 
semicircle cutting the perpendicular 
through B in D. BD will be the 
mean proportional required. (13.6.) 

105. Second Method.— Let AB and M*29. 
AC (Fig. 29) be the given lines. On 
AB describe a semicircle cutting the 
perpendicular at C in D. Join AD. 
AD is the mean proportional required. 
(Cor. 8.6.) Make AE = AD. 

Note. — This is a very convenient construction, and is often employed in the 
Division of Land. 




Fig. 30. 



Problem 13. — To divide a given line into parts having 
same ratio as two given numbers M and N". 

106. Let AB (Fig. 30) be the given 
line. Draw AC making any angle 
with AB. Lay off AD = M, taken 
from any scale of equal parts, and 
DE = N, taken from the same scale. 
Join BE, and draw DF parallel to it, 
and the thing is done. (2.6.) 



the 




CHAPTER III. 

PLANE TRIGONOMETRY. 



SECTION I. 
DEFINITIONS. 



107. Plane Trigonometry is the science which treats of 
the relations between the sides and angles of plane tri- 
angles ; which develops the principles by which, when 
any three of the six parts of a triangle, — viz. : the three 
angles and the three sides, — except the three angles, are 
given, the others may be found. It likewise treats of the 
properties of the trigonometrical functions. 

108. Measure of Angles. An angle is the inclination 
between two straight lines : it is measured by the inter- 
cepted arc of a circle described about the angular point as 
a centre. 

In the measurement of angles, it is not the absolute 
length of the arc that is needed, but the ratio which that 
length bears to the whole circumference. 

For the purpose of expressing this ratio readily, the cir- 
cumference is supposed to be divided into 360 parts, called 
degrees, each degree into 60 parts, called minutes, and 
each minute into 60 seconds. Degrees are marked with a 
cipher ° over them, minutes with one accent ', and seconds 
with two " . Thus, 37 degrees, 45 minutes, and 30 seconds, 
would be written 37° 45' 30". 

When we speak of an arc of 35°, we mean an arc which 

35 

is oFa of the circumference. An arc of 180° is half the 

45 



46 



PLANE TRIGONOMETRY. 



[Chap. III. 



Fig. 31. 



circumference, one of 90° is a quadrant, and of 45° the 
half of a quadrant. 

It is evident that, if several circles be described about 
the same point, the arcs intercepted between two lines 
drawn from the centre will bear the same ratio to the cir- 
cumferences of which they are portions. Thus, if around 
the point A (Fig. 31) two circles 
BCD and EFGr be described, cut- 
ting AK and AH in B, E, C, F, 
the arc BC will have to the cir- 
cumference BCD the same ratio 
as EF has to the circumference 
EFG. In the measurement of 
angles, it is a matter of indif- 
ference, therefore, what radius is 
assumed as that of the circle of reference. The radius 
which is generally adopted is unity. This value of the 
radius makes it unnecessary to write it down in the 
formulae. 

The radius adopted in the construction of the Table of 
Logarithmic Sines and Tangents, to be described hereafter, 
is 10,000,000,000. 




Fig. 32. 



109. The complement of an arc or 
angle is what it differs from a quad- 
rant, or 90°. Thus, DB (Fig. 32) is 
the complement of AB, and MD of 
AM. 




HO. The supplement of an arc or angle is what it wants 
of 180°. Thus, BE (Fig. 32) is the supplement of AB, and 
ME of AM. 



1U. Trigonometrical Functions. The trigonometri- 
cal functions are lines having definite geometrical relations 
to the arc to which they belcng. Those most in use are 
the sine, the cosine, the tangent, the cotangent, the secant, 
and the cosecant. 



Sec. I.] DEFINITIONS. 47 

The chord of an arc is the right line joining the extremi- 
ties of that arc. Thus, EM (Fig. 32) is the chord of the 
arc EM. 

The sine of an arc is the line drawn from one extremity 
of the arc, perpendicular to the diameter through the other 
extremity. BF (Fig. 32) is the sine of AB or of EB, and 
BL of BD. 

Note. — The sine of an arc is equal to the sine of its supplement. 

The cosine of an arc is the line intercepted between the 
foot of the sine and the centre. CF is the cosine of AB 
or of BE. 

Since CF = BL, it is manifest that the cosine of an arc 
is equal to the sine of its complement. 

The tangent of an arc is a line touching the arc at one 
extremity and produced till it meets the radius through 
the other extremity. Thus, AT is the tangent of AB, and 
DK of DB. 

The cotangent of an arc is the tangent of its complement. 
Thus, DK (Fig. 32) is the cotangent of AB. 

The secant of an arc is the line intercepted between 
the centre and the extremity of the tangent. Thus, CT 
(Fig. 32) is the secant of AB. 

The cosecant of an arc is the secant of the complement 
of that arc. Thus, CK (Fig. 32) is the cosecant of AB. 

The sine, cosine, &c. of an arc are also called the sine, 
cosine, &c. of the angle measured by that arc. Thus, BF 
and CF (Fig. 32) are the sine and cosine of the angle ACB. 

Note. — The tangent, cotangent, secant, or cosecant of an arc is equal to the 
tangent, cotangent, secant, or cosecant of its supplement. 

112. Properties of the Sines, Tangents, &c. of an 
arc or angle. 

The sine of 90°, the cosine of 0°, the tangent of 45°, the 
cotangent of 45°, the secant of 0°, and the cosecant of 90°, 
are each equal to radius. 

The square of the sine + the square of the cosine of 



48 PLANE TRIGONOMETRY. [Chap. III. 

any arc is equal to the square of radius. (Sin. 2 a + cos. 2 a 
= E 3 .) This is evident from the right-angled triangle CFB, 
(Fig. 32.) (47.1.) 

The square of the tangent -f the square of radius is equal 
to the square of the secant. Tan. 2 a -f- E 2 = sec. 2 a. (47.1.) 

Tan. a : E : : E : cotan. a, or tan. a. cot. a = E 2 . This is 
evident from the similarity of the triangles ACT and DKC, 
(Fig. 32,) which give (4.6) AT : AC : : CD : DK. J- 1 ; .' 

The sine of 30° and the cosine of 60° is each equal to 
half radius. 

113. Geometrical properties most employed in Plane 
Trigonometry. 

The angles at the base of an isosceles triangle are equal ; 
and conversely, if two angles of a triangle are equal, the 
sides which subtend them are equal. (5 and 6.1.) £ 

The external angle of a triangle is equal to the two 
opposite internal ones. (32.1.) 

The three interior angles of a triangle are equal to two 
right angles or 180°. (32.1.) 

Hence, if the sum of two angles be subtracted from 180°, 
the remainder will be the third angle. 

If one angle be subtracted from 180°, the remainder is 
the sum of the other angles. 

If one oblique angle of a right-angled triangle be sub- 
tracted from 90°, the remainder is the other angle. 

The sum of the squares of the legs of a right-angled tri- 
angle is equal to the square of the hypothenuse. (47.1.) 

The angle at the centre of a circle Fig. n. 

is double the angle at the circum- 
ference upon the same arc ; or, in 
other words, the angle at the cir- 
cumference of a circle is measured 
by half the arc intercepted by its 
sides. (20.3.) Thus, the angle ADB 
V is half ACB ; and is, therefore, mea- 
sured by one-half of the arc AB. 

The sides about the equal angles of equiangular tri- 
angles are proportionals. (4.6.) 



>kV.\ 




Sec. II.] DRAFTING OR PLATTING. 49 

SECTION II. 

DRAFTING OR PLATTING.* 

114. Drafting is making a correct drawing of the parts 
of an object. Platting is drawing the lines of a tract of 
land so as correctly to represent its boundaries, divisions, 
and the various circumstances needful to be recorded. It 
is, in fact, making a map of the tract. It is of great im- 
portance to a surveyor to be able to make a correct and 
neat plat of his surveys. The facility of doing so can only 
be acquired by practice; the student should, therefore, be 
required to make a neat and accurate draft of every pro- 
blem in Trigonometry he is required to solve, and of every 
survey he is required to calculate. It is not sufficient that 
he should draw a figure, as he does in his demonstrations in 
Geometry, that will serve to demonstrate his principles or 
afford him a diagram to refer to, but he should be obliged 
to make all parts in the exact proportion given by the data, 
so that he can, if needful, determine the length of any line, 
or the magnitude of any angle, by measurement. 

115. Straight lines. Straight lines are generally drawn 
with a straight-edged ruler. If a very long straight line is 
needed, a fine silk thread may be stretched between the 
points that are to be joined, and points pricked in the 
paper at convenient distances; these may then be joined 
by a ruler. 

In drawing straight lines, care should be taken to avoid 
determining a long line by producing a short one, as any 
variation from the true direction will become more mani- 
fest the farther the line is produced. When it is necessary 
to produce a line, the ruler is fixed with most ease and cer- 
tainty by putting the points of the compasses into the line 
to be produced, and bringing the ruler against them. 

116. Parallels. Parallels may be drawn as described in 



* Various hints in this section have been derived from Gillespie's "Laud 
Surveying." . 



50 



PLANE TRIGONOMETRY. 



[Chap. Ill 




Arts. 97, 98. Practically, however, it is better to draw 
them by some instrument specially adapted to the purpose. 

The square and ruler are very convenient instruments 
for this purpose. The square consists of two arms, which 
should be made at right angles to each other, to facilitate 
the erection of perpendi- Fig. 33. 

culars. Let AB (Fig. 33) be 
the line to which a parallel 
is to be drawn through C. 
Adjust one edge of the 
square to the line AB, and 
bring a ruler firmly against 
the other leg; move the 
square along the ruler un- 
til the edge coincides with 
C : this edge will then be 
parallel to the given line. 

If a T square be substituted for a simple right angle, it 
may be held more firmly against the ruler. 

Instead of a square, a right-angled triangle is frequently 
used. The legs should 
be made accurately at 
right angles, that it may 
be used for drawing per- 
pendiculars. Let AB 
(Fig. 34) be the line, and 
C the point through which 
it is required to draw a 
parallel. Bring one edge of the triangle accurately to the 
line, and then place a ruler against one of the other sides. 
Slide the triangle along the ruler until the point C is in 
the side which before coincided with the line : this side is 
then parallel to the given line. 

The parallel rulers which accompany most cases of in- 
struments are theoretically accurate. They are, however, 
generally made with so little care that they cannot be de- 
pended on where correctness is required ; and, even if made 
true, they are liable to become inaccurate in consequence 
of wear of the joints. 



Fig. 34. 
C 




Sec. II.] 



DRAFTING OR PLATTING. 



51 



Fig. 35. 




117. Perpendiculars. Perpendiculars may be drawn as 
directed, (Art. 88, et seq.) A more ready means is to place 
one leg of the square (Fig. 33) upon the line : the other will 
then be perpendicular to that line. The triangle is another 
very convenient instrument for this 
purpose. Let AB (Fig. 35) be the 
line to which a perpendicular is to 
be drawn. Place the hypothenuse 
of the triangle coincident with AB, 
and bring the ruler against one of 
the other sides. Remove the tri- 
angle and place it with the third 
side against the ruler, as at D : then the hypothenuse will be 
perpendicular to AB. 

This method requires the angle of the triangle to be pre- 
cisely a right angle. To test 
whether it is so, bring one leg 
against a ruler, as at A, (Fig. 36,) 
and scribe the other leg. Reverse 
the triangle, and bring the right 
angle to the same point A, and a 

again scribe the leg. If the angle is a right angle, the two 
scribes will exactly coincide. If they do not coincide, the 
triangle requires rectification. 




118. Circles and Arcs. These are generally drawn with 
the compasses, which should have one leg movable, so that 
a pen or a pencil may be inserted instead of a point. 
When circles of long radii are required, the beam compasses 
should be used. 

These consist of a bar of wood or metal, dressed to a 
uniform size, and having two slides furnished with points. 
These slides can be adjusted to any part of the beam, and 
clamped, by means of screws adapted to the purpose. The 
point connected with one of the slides is movable, so that a 
pencil or drawing pen may be substituted. 

When the beam compasses are not at hand, a strip of 
drawing paper or pasteboard may be substituted : a pin 
through one point will serve as a centre; the pencil 



52 



PLANE TRIGONOMETRY. 



[Chap. III. 



point can be passed through a hole at the required 
distance. 

119. Angles. Angles may be laid off by a protractor. 
This is usually a semicircle of metal, the arc of which is 
divided into degrees. To use it, place it with the centre at 
the point at which the angle is to be made, and the straight 
edge coincident with the given line ; then with a fine point 
prick off the number of degrees required, and join the point 
thus determined to the centre. 

The figures on the protractor should begin at each end 
of the arc, as represented in Fig. 3T. 

Fig. 37. 




120. By the Scale of Chords. The scale of chords, 
which is engraved on the ivory scales contained in a box 
of instruments, may also be used for making angles. For 
this purpose take from the scale the chord of 60° for a 
radius. "With the point A, at which the angle is to be made, as 
a centre, and that radius, describe an arc. Take off from 
the scale the chord of the required number 
of degrees and lay it on the arc from the 
given line, join the extremity of the arc 
thus laid off to the centre, and the thing is 
done. 

Thus, if at the point A (Fig. 38) it were 
required to make an angle BAC of 47°. 



Fie. 38. 



Sec. II. ] DRAFTING OR PLATTING. 53 

"With, the centre A and radius equal to the chord of 60° 
describe the arc BC. Then, taking the chord of 47° from 
the scale, lay it off from B to C. Join AC, and BAC will 
be the required angle. 

If an angle of more than 90° is required : first lay off 90°, 
and from the extremity of that arc lay off the remainder. 

121. By the Table of Chords. The table of chords 
(page 97 of the tables) affords a much more accurate means 
of laying off angles. 

Take for a radius the distance 10 from any scale of equal 
parts, — to be described hereafter, — and describe the arc BC, 
(Fig. 38.) Then, finding the chord of the required angle 
by the table, multiply it by 10, and, taking the 'product 
from the same scale, lay it off from B to C as before. Join 
AC, and the thing is done. 

If the angle is much over 60° it is best to lay off the 60° 
first. This is done by using the radius as a chord. The 
remainder can then be laid off from the extremity of the 
arc of 60° thus determined. 

122. Distances. Every line on a draft should be drawn 
of such a length as correctly to represent the distance of 
the points connected, in due relation to the other parts of 
the drawing. In perspective drawing, the parts are deline- 
ated so as to present to the eye the same relations that 
those of the natural object do when viewed from a particular 
point. To produce this effect the figure must be distorted. 
Bight angles are represented as right, obtuse, or acute, ac- 
cording to the position of the lines ; and the lengths of lines 
are proportionally increased or diminished according to 
their position. In drafting, on the contrary, every part 
must be represented as it is. The angles should be of the 
same magnitude as they are in reality, and the lines should 
bear to each other the exact ratio that those which they 
are intended to represent do. The plat should, in fact, be 
a miniature representation of the figure. 

123. Drawing to a Scale. In order that the due pro- 



54 PLANE TRIGONOMETRY. [Chap. III. 

portion should exist in the parts of the figure, every line 
should be made some definite part of the length of that 
which it is intended to represent. This is called drawing to 
a scale. The scale to be used depends on the size of the 
map or draft that is required, and the purposes for which it 
is to be used. Carpenters often use the scale of an inch to 
a foot : the lines will then be the twelfth part of their real 
length. In plats of surveys, or maps of larger tracts of 
country, a greater diminution is necessary. The scale 
should, however, in all cases, be adapted to the purpose 
intended and to the number of objects to be represented. 
Where the purpose is merely to give a correct representa- 
tion of the plat, without filling up the details, the main 
object will be to make the map of a convenient size; but 
where many details are to be represented the scale should 
be proportionally larger. 

Thus, for example, in delineating a harbor where there 
are few obstructions to navigation, a map on a small scale 
may be drawn ; but where the rocks and shoals are nume- 
rous, the scale should be so large that every part may be 
perfectly distinct. 

The scales on which the drawing is made should always 
be mentioned on the map. They may be expressed by 
naming the lengths which are used as equivalents, thus, — 
" Scale, 10 feet to an inch, 1 mile to an inch, 3 chains to 
a foot;" or better fractionally, thus, — 1 : 100, 1 : 250, 
1 : 10,000, &c. 

124. Surveys of Farms. Where the farm is small, 1 
chain* to an inch, (1 : 792,) or 2 chains to the inch, (1 : 1584,) 
may be used ; but if the tract be large, as this would make 
a plat of a very inconvenient size, a smaller scale must be 
adopted. When, however, any calculations are to be based 
on measurements taken from the plat, a smaller scale than 
3 chains to the inch (1 : 2376) should not be employed. 



* The surveyor's chain — commonly called Guntcr's Chain — is 4 poles, or 66 
feet, in length, and is divided into one hundred links, each of which is therefore 
.66 feet, or 7.92 inches in length. 



Sec. II.] 



DRAFTING OR PLATTING. 



55 



125. Scales, Scales are generally made of ivory or box- 
wood, having a feather-edge, on which the divisions are 
marked. The distances can then be laid off by placing the 
ruler on the line, and pricking the paper or marking it with 
a fine pointed pencil; or the length of a line may be read 
off without any difficulty. Boxwood scales, if the wood is 
clear from knots, are to be preferred to ivory. They are 
less liable to warp, and suffer less expansion and con- 
traction from changes in the hygrometric condition of the 
atmosphere. 

Paper scales are often employed. These may be pro- 
cured with divisions to suit almost any purpose, or the sur- 
veyor may make them himself. Take a piece of drawing- 
paper, and cut a slip about an inch in width ; draw a line 
along its middle, and divide it as desired, either into inches 
or tenths of a foot. The end division should be subdivided 
into ten parts, and perpendiculars drawn through all the 
divisions, as represented in the figure, (Fig. 39.) Each of 
these parts may then represent a chain, ten chains, &c. 

Fig. 39. 



















! 










\ 


















i 








.. 


s 



Paper scales, being subject to nearly the same expansion 
and contraction as the paper on which the map is drawn, 
are, on this account, preferable to those made of wood or 
ivory. They cannot, however, be divided with the same 
accuracy. 



126. The plane diagonal scale (Fig. 40) consists of eleven 

Fig. 40. 



Q 



A 2 4 6\ 8 B 




i D 



56 PLANE TRIGONOMETRY. [Chap. in. 

lines drawn parallel and equidistant. These are crossed at 
right angles by lines 1, 2, 3, drawn usually at intervals of 
half an inch. The first division, on the upper and lower 
lines, is subdivided into ten equal parts : diagonal lines are 
then drawn, as in the figure, from each division of the top 
to the next on the bottom, — the first, from A to the first 
division on the bottom line ; the second, from the first on 
the top to the second on the bottom ; and so on. 

It is evident that, whatever distance the primary division 
from A to 1, or 1 to 2, &c. represents, the parts of the line 
AB will represent tenth parts of that distance. If then 
it were required to take ofT the distance of 47 feet on a 
scale of half an inch to 10 feet, the compasses should be 
extended from E to F. 

The diagonal lines serve to subdivide each of the smaller 
divisions into tenths, thus : — The first diagonal, extending 
from A to the first division on the bottom line and crossing 
ten equal spaces, will have advanced ^ of one of those 
divisions at the first intermediate line, $ at the second, ^ at 
the third, and so on. All the other diagonals will advance 
in the same manner. 

If then the distance were taken from the line AC along 
the horizontal line marked 6 to the fourth diagonal, the 
distance would be .46, the division AB being a unit, or 
4.6 if AB were 10. To take off, then, 39.8 feet on a scale 
of half an inch to 10 feet, the compasses should be ex- 
tended to the points marked by the arrow heads G and H : 
similarly, 46.7, on the same scale, would extend from one 
of the arrow heads on the seventh line to the other. 

In using the diagonal scale the primary divisions should 
always be made to represent 1, 10, 100, or 1000. When any 
other scale is required, — say 1 : 300, — it is better to divide 
or multiply all the distances and then take off the results. 
Thus, if 83.7 were required to be taken off on a scale of 
J inch to 30 feet, first divide 83.7 by 3, giving 27.9, and 
then take off the quotient on a scale of J- inch to 10 feet. 
The other lines must all be reduced in the same proportion. 
The above method requires less calculation, and involves 



Sec. II.] 



DRAFTING OR PLATTING. 



57 



less liability to error, than that of determining the value of 
each division on the reduced scale. 

127. Proportional Scale. On most of the rulers fur- 
nished with cases of instruments there is another set of 
scales, divided as below, (Fig. 41.) 















Fig. 


41. 










1 60 


"in . 

[ran™ J- 


2 


3 


' 4f 5 


6| 7I s| 9| a |p 1 


2 


3| 4 


5| 6| 7 




50 


m_ it. »i si * 


5] 6| r.l s| 9 


l|o l| 2| s| 4 


- 


45 


■.V'l'M'J 


l| 2,] . Z.\ 4-! 5| 6[ 7|' 1 8 


9t . lid ll 2I 3l 


ii)lJ!J||] - 


40 


njff|tt|fli 


ll sl 3J 4,| 5l. _ 6.1 . 7 


sl . 9|< i|o l\ 




35 




ij Z\ Z\ 4r\ b\ 6 


7 1 & 1 9 1 aio 


IJ II ! 11 1 1 1 


'• 1 ° 1 "1. - L l u 


40 


Li.juj.'iji. 




2 


3, 


4 


5 


*4 T »4 1 


Ill 1 |-ll|| 















The figures on the left express the number of divisions to 
the inch. To lay off 97 feet on a scale of 40 feet to the 
inch, the compasses would be extended between the arrow- 
heads on the line 40. Scales of this kind are very con- 
venient in altering the size of a drawing. Suppose, for 
example, it is desired to reduce a drawing in the ratio of 5 
to 3 : the lengths of the lines should be determined on the 
scale marked 30, and the same number of divisions on the 
scale 50 will give a line of the desired length. 

128. Vernier Scale. Make a scale (Fig. 42) with inches 
divided into tenths, and mark the end of the first inch 0, 
of the second 100, and so on. From the zero point, back- 
wards, lay off a space equal to eleven tenths of an inch, 
and divide it into ten equal parts, numbering the parts 
backwards, as represented in the figure. This smaller scale 



Fig. 42. 



100 



2100 



i in it 

88 66 144 



II 



is a vernier. ISTow, since the ten divisions of the vernier 
are equal to eleven of the scale, each of the vernier divisions 



58 PLANE TRIGONOMETRY. [Chap. III. 

is equal to Q of ^ = ^ of an inch. From the zero point, 
therefore, to the second division of the vernier is .22 inch, 
to the third .33, and so on. 

To measure any line by the scale, take the distance in 
the compasses, and move them along the scale until you 
find that they exactly extend from some division on the 
vernier to a division on the scale. Add the number 
on the scale to the number on the vernier for the dis- 
tance required. Thus, suppose the compasses extended 
from 66 on the vernier to 110 on the scale, the length is 
176. 

To lay off a distance by the scale, for example 175, take 
55 from 175, and 120 is left : extend the compass from 120 
on the scale to b^> on the vernier. To lay off 268 = 180 + 
88, extend the compasses from 180 on the scale to 88 on the 
vernier, as marked by the arrow heads. 

The vernier scale is equally accurate with the diagonal 
scale, and much more readily made. 



SECTION III. 

TABLES OF TRIGONOMETRICAL FUNCTIONS. 

129. Table of Natural Sines and Cosines, This table 
(page 87 of the Tables) contains the sines and cosines to five 
decimal places for every minute of the quadrant. The 
table is calculated to the radius 1. As the sine and cosine 
are always less than radius, the figures are all decimals. In 
the table the decimal point is omitted. If the sine and 
cosine is wanted to any other radius, the number taken 
from the table must be multiplied by that radius. 

To take out the sine or cosine of an arc from this table, 
look for the degrees, if less than 45, at the top of the table, 
and for the minutes at the left ; then, in the column headed 
properly, and opposite the minutes, will be the function 
required. If the degrees are 45 or upwards they will be 



Sec. III.] TRIGONOMETRICAL FUNCTIONS. 59 

found at the bottom, and the minutes at the right. The 
name of the column is at the bottom. 

Thus, the sine of 32° 17', found under 32° and opposite 
17', is .53411. 

The cosine of 53° 24', found over 53° and opposite 24' in 
the right-hand column, is .59622. 

130. The table of natural sines and cosines is of but little 
use in trigonometrical calculations, these being generally 
performed by logarithms. It is principally employed in 
determining the latitudes and departures of lines. 

131. Table of Logarithmic Sines, Cosines, &c. This 
table contains the logarithms of the sines, cosines, tangents, 
and cotangents, to every minute of the semicircle, the radius 
being 10 000 000 000 and its logarithm 10. The logarithmic 
sine of 90°, cosine of 0°, tangent of 45°, and cotangent of 
45°, is each 10. 

The sine, cosine, tangent, and cotangent, of every arc being 
equal to the sine, cosine, tangent, and cotangent, of its supple- 
ment, and also to the cosine, sine, cotangent, and tangent, of its 
complement, the table is only extended to forty five pages, 
the degrees from to 44 inclusive being found at the top, 
those from 45 to 135 at the bottom, and from 136 to 180 at 
the top. The minutes are contained in the two outer 
columns, and agree with the degrees at the top and bottom 
on the same side of the page. 

The columns headed Diff. 1" contain the difference of 
the function for a change of V in the arc. These differ- 
ences are calculated by dividing the differences of the suc- 
cessive numbers in the columns of the functions by 60. By 
an inspection of these columns of difference it will be seen 
that, except in the first few pages, they change very slowly. 
In these, in consequence of the rapid change of the func- 
tion, the differences vary very much. The difference set 
down will not, therefore, be accurate, except for about the 
middle of the minute. The calculations for seconds, there- 
fore, are not in these cases to be depended on. To obviate 
this inconvenience, and give to the first few pages a degree 



60 PLANE TRIGONOMETRY. [Chap. III. 

of accuracy commensurate with that of the rest of the table, 
the sines and tangents are calculated to every 10 seconds, 
and these are the same as the cosines and cotangents of 
arcs within two degrees of 90.* 

132. Use of Table. To take out any function from the 
table, seek the degrees, if less than 45° or more than 135°, at 
the top of the page, and the minutes in the column on the 
same side of the page as the degrees. Then, in the proper 
column, (the title being at the top,) and opposite the minutes, 
will be found the value required. 

If the degrees are between 45° and 135°, seek them at 
the bottom of the page, the minutes being found, as before, 
at the same side of the page as the degrees. The titles of 
the columns are also at the bottom. 

Examples. 

Ex. 1. Required the sine of 37° 17'. Ans. 9.782298. 

Ex. 2. Required the cosine of 127° 43'. Ans. 9.786579. 

Ex. 3. Required the cotangent of 163° 29'. 

Ans. 10.527932. 

Ex. 4. Required the tangent of 69° 11'. 

Ans. 10.419991. 

133. If there are seconds in the arc, take out the function 
for the degrees and minutes as before. Multiply the num- 
ber in the difference column by the number of seconds, and 
add the product to the number first taken out, if the func- 
tion is increasing, but subtract, if it is decreasing : the 
result will be the value required. 

If the arc is less than 90° the sine and tangent are in- 
creasing, and the cosine and cotangent are decreasing ; but if 
the arc is greater than 90° the reverse holds true. 

* The rectangle of the tangent and cotangent of an arc being equal to the 
square of radius, their logarithms are arithmetical complements (to 20) of each 
other. Our column of differences serves for both these functions. It is placed 
between them. 



Sec. III.] TRIGONOMETRICAL FUNCTIONS. 61 

Ex. 1. What is the tangent of 37° 42' 25"? 

The tangent of 37° 42' is 9.888116 

Diff. 1" 4.35 

25 

2175 
87 



Diff. 25" 108.75 + 109 

Tangent 37° 42' 25" 9.888225 

Ex. 2. What is the cosine of 129° 17' 53"? 

The cosine of 129° 17' is 9.801511 

Diff. 1" 2.57 

53 

7 71 

128 5 



Diff. 53" 136.21 +136 

Cosine 129° 17' 53" 9.801647 

Ex. 3. What is the sine of 63° 19' 23"? 

Ans. 9.951120. 

Ex. 4. What is the cosine of 57° 28' 37"? 

Ans. 9.730491. 

Ex. 5. What is the tangent of 143° 52' 16"? 

Ans. 9.863314. 

Ex. 6. What is the sine of 172° 19' 48"? 

Ans. 9.125375. 

If the sine or tangent of an arc less than 2° or more 
than 178°, or the cosine or cotangent of an arc between 
88° and 92°, is required, it should be taken from the first 
pages of the table. Take out the function to the ten 
seconds next less than the given arc, multiply one tenth of 
the difference between the two numbers in the table by the 
odd seconds, and add or subtract as before. 

The cotangent of an arc less than 2° may be found by 
taking out the tangent, and subtracting it from 20.000000 ; 
so likewise the tangent of an arc between 178° and 180° 
is found by taking the complement to 20.000000 of its 
cotangent. 



62 PLANE TRIGONOMETRY. [Chap. III. 

Ex. 1. Eequired the sine of 1° 27' 36". 

Sine of 1° 27' 30" is 8.405687 

^ of difference 82.6 

6 

Difference 6" 495.6 496 

Sine of 1° 27' 36" 8.406183 

Ex. 2. What is the cosine of 88° 18' 48"? 

Ans. 8.468844. 

Ex. 3. What is the sine of 179° 19' 13"? 

Ans. 8.074198. 

134. To find the Arc corresponding to any Trigo- 
nometric Function. 

If degrees and minutes only be required, seek, in the pro- 
per column, the number nearest that given ; and if the title 
is at the top the degrees are found at the top, and the minutes 
under the degrees; but if the title is at the bottom the 
degrees are at the bottom, and the minutes on the same side 
as the degrees. 

If seconds are desired, seek for the number corresponding 
to the minute next less than the true arc, and take the 
difference between that number and the given one : divide 
said difference by the number in the difference column, for 
the seconds. 

Ex. 1. What is the arc whose sine is 9.427586 ? 

9.427586 
Sine of 15° 31' is 9.427354 

7.58)232.00(31" 

227 4 



4.60 



The arc is, therefore, 15° 31' 31". 



Sec. III.] TRIGONOMETRICAL FUNCTIONS. 63 

Ex. 2. What is the arc whose cotangent is 10.219684? 

10.219684 
Cotangent of 31° 5' is 10.219797 

4.76)113.00(23.7" 
952 

17 80 
14 28 



3.52 



The arc is, therefore, 31° 5' 24". 



Ex. 3. Required the arc the cosine of which is 9.764227. 

Ans. 54° 28' 27". 

Ex. 4. Required the arc the tangent of which is 
10.876429. Ans. 82° 25' 44". 

Ex. 5. What is the arc the cotangent of which is 11.562147? 

As this corresponds to an arc less than 2°, take it from 
20.000000 : the remainder, 8.437853, is the tangent. The 
arc is found as follows : — 

8.437853 
1° 34' 10" tang. 8.437732 

Diff. tol" 76.8 ) 121.0(1.6" 

76 8 

44.20 

The angle is, therefore, 1° 34' 11.6". 

Ex. 6. What arc corresponds to the cotangent 8.164375? 

Ans. 89° 9' 48.6". 

135. Table of Chords. This table contains the chords 
of arcs to 90° for every 5 minutes. It is principally used 
in laying off angles, as explained in Art. 120, and in pro- 
tracting surveys by the method of Art. 343. 






54 



PLANE TRIGONOMETRY. 



[Chap. III. 



SECTION IV. 

ON THE NUMERICAL SOLUTION OF TRIANGLES. 

136. Definition. The solution of a triangle is the deter- 
mination of the numerical value of certain parts when 
others are given. To determine a triangle, three inde- 
pendent parts must be known, — viz. : either the three sides, 
or two sides and an angle, or the angles and one side. The 
three angles are not of themselves sufficient, since they are 
not independent, — any one of them being equal to the dif- 
ference between the sum of the others and 180°. 

In the solution of triangles several cases may be distin- 
guished ; these will be treated of separately. These cases 
are applicable to all triangles. But as there are special 
rules for right-angled triangles, which are simpler than the 
more general ones, they will first be given. 



A.— THE NUMERICAL SOLUTION OF RIGHT-ANGLED 

TRIANGLES. 

137. The following rules contain all that is necessary for 
solving the different cases of right-angled triangles. 

1. The hypothenuse is to either leg as radius is to the sine of 
the opposite angle. 

2. The hypothenuse is to one leg as radius is to the cosine of 
the adjacent angle. 

3. One leg is to the other as radius is to the tangent of the angle 
adjacent to the former. 

Demonstration. — Let ABC (Fig. 43) be a Fig. 43. 

triangle right-angled at B. Take AD any ra- 
dius, and describe the arc DE ; draw EF and 
DG perpendicular to AB. Then EF will be the 
sine, AF the cosine, and DG the tangent, of 
the angle A. Now, from similar triangles we 
have — 




1. AC 

2. AC 

3. AB 



CB 
AB 
BC 



AE 
AE 
AD 



EF 
AF 

DG 





A 






r : 


sin. 


A. 


Rule 1 ; 


r 


cos. 


A. 


Rule 2 ; 


r 


tan. 


A. 


Rule 3. 



¥ D 



Sec. IV.] NUMERICAL SOLUTION OF TRIANGLES. 65 

Examples. 
Ex. 1. In the triangle ABC, right-angled at B, there are 
given the base AB = 57.23 chains, and the angle A 35° 27' 
25", to find the other sides. 

Construction. 
Make AB (Fig. 44)= 57.23, taken Fig. u. 

from a scale of equal parts. At the 
point A make the angle BAC = 
35° 27'. Erect the perpendicular 
BC, meeting AC in C, and ABC 
is the triangle required. 






Calculation. 








Eule 3. r : tan. A : : AB 


:BC. 




Eule 2. cos. A : r : : AB 


: AC. 




For facility of calculation, the proportions are generally 


written vertically, 


as below. 








As rad. 




log. 


10.000000 




: tan. A 


35° 27' 25" 




9.852577 




:: AB 


57.23 ch. 




1.757624 




: BC 


40.76 




1.610201 




As cos. A 


35° 27' 25" Ar. Co. 




0.089081 




: rad. 






10.000000 




:: AB 


57.23 




1.757624 




: AC 


70.26 




1.846705 





Ex. 2. Given AB = 47.50 chains, and AC = 63.90 chains, 
to find the angles and side BC. 

Eule 2. 

As AC , 63.90 Ar. Co. 8.194499 

: AB 47.50 1.676694 

:: rad. 10.000000 

: cos. A 41° 58' 57" 9.871193 
90 

C 48° 1' 3" 
5 



66 





PLANE TRIGONOMETRY. 


[Cn 




EULE 1. 




As rad. 




10.000000 


: sin. A 


41° 58' 57" 


9.825363 


:: AC 


63.90 


1.805501 


: CB 


„ 42.74 


1.630864 



Ex. 3. Given the two legs AB = 59.47 yards, and BC = 
48.52 yards, to find the hypothenuse and the angles. 
Ans. A 39° 12' 36", C 50° 47' 24", and AC 76.75 yds.- 

Ex. 4. Given the hypothenuse AC = 97.23 chains, the 
perpendicular BC = 75.87 chains, to find the rest. 

Ans. A 51° 17' 22", C 38° 42' 38", AB 60.81 ch. 

Ex. 5. Given the angle A = 42° 19' 24", and the perpen- 
dicular BC = 25.54 chains, to find the other sides. 

Ans. AC 37.932 ch., AB 28.045 ch. 

Ex. 6. Given the angle C = 72° 42' 9", and the hypo- 
thenuse AC = 495 chains, to find the other sides. 

Ans. AB 472.612 ch., BC 147.18 ch. 

Ex. 7. In the right-angled triangle ABC we have the 
base AB = 63.2 perches, and the angle A 42° 8' 45", to 
find the hypothenuse and the perpendicular. 

Ans. BC 57.20 p., AC 85.24 p. 

138. When two sides are given, the third may be found 
by (47.1) ; thus, 

1. Given the hypothenuse and one leg, to find the other. 

Rule. From the square of the hypothenuse subtract the square 
of the given leg : the square root of the remainder will be the 
other leg ; or, 

Multiply the sum of the hypothenuse and given leg by their 
difference : the square root of this product will be the other leg. 

This is evident from (47.1) and (cor. 5.2.) 

2. Given the two legs, to find the hypothenuse. 

Rule. Add the squares of the two legs, and extract the square 
root of the sum : the result will be the hypothenuse. 



Sec. IV.] 



NUMERICAL SOLUTION OF TRIANGLES. 



67 



Examples. 
Ex. 1. Given the hypothenuse AC = 45 perches, and the 
leg BC = 29 perches, to find the other leg. 



Rule 1. AB = n/AC 2 - BC 2 = ^2025 - 841 = s/1184 = 
34.41. 



or, 



AB = •(AC + BC).(AC - BC) = x/74 x 16 = 
•1184 = 34.41. 

Ex. 2. The two legs AB and AC are 6 and 8 respectively : 
what is the hypothenuse ? Ans. 10. 

Ex. 3. The hypothenuse AC is 47.92 perches, and the 
leg AB is 29.45 perches : required the length of BC. 

Ans. 37.8 perches. 

Ex. 4. The hypothenuse of a right-angled triangle is 
49.27 yards, and the base 37.42 yards : required the perpen- 
dicular. Ans. 32.05. 



B — THE NUMERICAL SOLUTION OF OBLIQUE-ANGLED 

TRIANGLES. 

CASE 1. 

139. The angles and one side, or two sides and an angle oppo- 
site to one of them, being given, to find the rest 

Rule. 

1. As the sine of the angle opposite the given side is to the 
sine of the angle opposite the required side, so is the given side 
to the required side. 

2. As the side opposite the given angle is to the other given 
side, so is the sine of the angle opposite to the former to the sine 
of the angle opposite the latter. 

Demonstration. — Both the above rules are combined in the general propo- 
sition. The sides are to one another as the sines of their opposite angles. 



Let ABC (Fig. 45) be any triangle. From C let fall 
CD perpendicular to AB. Then (Art. 137) AC : CD : : r 
: sin. A, and CD : CB : : sin. B : r. Whence (23.5) AC : 
CB : : sin. B : sin. A. 




68 



PLANE TRIGONOMETRY. 



[Chap. III. 



Examples. 

Ex. 1. In the triangle ABC are given AB = 123.5, the 
angle B = 39° 47' 20", and C = 74° 52' 10": required the 
rest. 

Construction. 

The angle A = 180 - (B + C) = 180° - 114° 39' 30" = 
65° 20' 30". 

Draw AB (Fig. 45) = 123.5. At the points A and B 
draw AC, BC, making the angles BAC and ABC equal, 
respectively, to 65° 20' 30" and 39° 47' 20" ; then will ABC 
be the triangle required. 

Calculation. 



As sin. C 


74° 52' 10" 


A. C. 0.015322 


: sin. B 


39° 47' 20" 


9.806154 


:: AB 


123.5 


2.091667 


: AC 


81.87 


1.913143 


As sin. C 




A. C. 0.015322 


: sin. A 


65° 20' 30" 


9.958474 


:: AB 




2.091667 


: BC 


116.27 


2.065463 



Ex. 2. Given the side AB = 327, the side BC = 238, and 
the angle A = 32° 27', to determine the rest. 



Construction. 

Make AB (Fig. 46) = 327 ; and 
at the point A draw AC making 
the angle A = 32° 47'. With the 
centre B and radius = 238 describe 
an arc cutting AC in C ; then will 
ABC be the triangle required. 



Fig. 46. 






Calculation. Bule 2. 


AsBC 


238 A. C. 7.623423 


: AB 


327 2.514548 


: : sin. A 


32° 47' 9.733569 


: sin. C 


48° 4' 6" 9.871540 


or 


131° 55' 54" 



Sec. IV.] NUMERICAL SOLUTION OF TRIANGLES. 69 





C acute. 








As sin. C 


48° 4' 6" 


A. 


C. 


0.128460 


: sin. B 


99° 8' 54" 






9.994441 


:: AB 


327 






2.514548 


: AC 


433.97 

C obtuse. 






2.637459 


As sin. C 


131° 55' 54" 


A. 


c. 


0.128460 


: sin. B 


15° 17' 6" 






9.420979 


:: AB 








2.514548 


: AC 


115.87 






2.063987 



Note. — It "will be seen that in the above example the result is uncertain. 
The sine of an angle being equal to the sine of its supplement, it is impossible, 
from the sine alone, to determine whether the angle should be taken acute or 
obtuse. By reference to the construction, (Fig. 46,) we see that whenever the 
side opposite the given angle is less than the other given side, and greater than 
the perpendicular BD, the triangle will admit of two forms : ABC, in which 
the angle opposite to the side AB is acute, and ABC 7 , in which it is obtuse. 
If BC were greater than BA, the point C' would fall on the other side of A, 
and be excluded by the conditions. If it were less than BD, the circle would 
not meet AC, and the question would be impossible. 

Ex. 3. Given the side AB 37.25 chains, the side AC = 
42.59 chains, and the angle C 57° 29' 15", to determine 
the rest. 

Ans. BC 32.774 chains, A = 47° 53' 52", and B = 74° 
36' 53". 

Ex. 4. Given the angle A 29° 47' 29", the angle B = 24° 
15' 17", and the side AB 325 yards, to find the other sides. 

Ans. AC = 164.93, BC = 199.48. 

Ex. 5. The side AB of an obtuse-angled triangle is 
127.54 yards, the side AC 106.49 yards, and the angle 
B 52° 27' 18", to determine the remaining angles and the 
side BC. 

Ans. C = 108° 16' 3", A = 19° 16' 39", BC = 44.34. 

Ex. 6. Given AB = 527.63 yards, AC = 398.47 yards, 
and the angle B 43° 29' 11", to determine the rest. 

Ans. C = 65° 40' 44", A = 70° 50' 5", BC = 546.93; 
or, C = 114° 19' 16", A = 22° 11' 33", BC = 218.71. 



70 PLANE TRIGONOMETRY. [Chap. Ill 

CASE 2. 

140. Two sides and the included angle being given, to determine 
the rest. 

BULE 1. 

Subtract the given angle from 180° : the remainder will be the 
sum of the remaining angles. Then, 

As the sum of the given sides is to their difference, so is the 
tangent of half the sum of the remaining angles to the tangent 
of half their difference. 

This half difference added to the half sum will give the angle 
opposite the greater side, and subtracted from the half sum will 
give the angle opposite the less side. 

Then having the angles, the remaining side may be found by 
Case 1. 

Demonstration. — The second paragraph of this rule may be enunciated in 
general terms ; thus, 

As the sum of two sides of a pla?ie triangle is to Fig. 47. 

their difference, so is the tangent of half the sum of 
the angles opposite those sides to the tangent of half 
the difference of those angles. 

Let ABC (Fig. 47) be the triangle of which the 
side AC is greater than AB. With the centre A 
and radius AC describe a circle cutting AB pro- 
duced in E and F. Join EC and CF, and draw 
FG parallel to BC. Then, because ABC and AFC 
have the common angle A, AFC -j- ACF = ABC 
4-ACB. Whence AFC = J (ABC -f AC B) ; and, 

since the half sum of two quantities taken from the greater leaves their half 
difference, CFG = EFG — EFC = ABC — EFC = £ (ABC — ACB). 

Now, since the angle ECF is an angle in a semicircle, it is a right angle. 
Therefore, if with the centre F and radius FC an arc be described, EC and 
CG will be the tangents of EFC and CFG, or of the half sum and half dif- 
ference of ABC and ACB. But (2.6) EB : BF : : EC : CG. 
Whence AC -f AB : AC — AB : : tan. £ (ABC + ACB) : tan. £ (ABC — ACB). 

Examples. 

Ex. 1. Given AB = 527 yards, AC = 493 yards, and the 
angle A =37° 49'. 

Here C + B = 180° - 37° 49' = 142° 11', and 




Sec. IV.] NUMERICAL SOLUTION OF TRIANGLES. 71 



As AB + AC 


1020 


A.C. 6.991400 


: AB-AC 


34 


1.531479 


C+B 
::tan. g 


71° 5' 30" 
5° 33' 29" 


10.465290 


C-B 

: tan. — - — 

2 


8.988169 


C 


76° 38' 59" 




B 


65° 32' 1" 




As sin. C 


76° 38' 59" 


A.C. 0.011897 


: sin. A 


37° 49' 


9.787557 


::AB 


527 


2.721811 


: BC 


332.10 


2.521265 



Ex. 2. In the triangle ABC are given AB = 1025.57 yaids, 
BC = 849.53 yards, and the angle B = 65° 43' 20", to find 
the rest. 

Ans. A = 48° 52' 10", C = 65° 24' 30", AC = 1028.13. 

Ex. 3. Two sides of a triangle are 155.96 feet and 
217.43 feet, and their included angle 49° 19', to find the 
rest. 

Ans. Angles, 85° 4' 12", 45° 36' 48", side, 165.49. 

Eule 2. 

141. As the less of the two given sides is to the greater, so is 
radius to the tangent of an angle; and as radius is to the tangent 
of the excess of this angle above 45°, so is the tangent of the 
half sum of the opposite angles to the tangent of their half 
difference. 

Having found the half difference, proceed as in Rule 1. 

Note. — This rule is rather shorter than the last, where the two sides have 
been found in a preceding calculation, and thus their logarithms are 
known. 




,2 PLANE TRIGONOMETRY. [Chap. III. 

Demonstration. — Let ABC (Fig. 48) be any p ^ig. 48. 

plane triangle. Draw BD perpendicular to AB, the 
greater, and equal to BC, the less side. Make BE = 
BD, and join ED. Then, since BE = BD, the angle 
BED = BDE ; and since EBD is a right angle, BDE 
= 45°. But BED + BDE = 2 BDE = BAD -f 
BDA, and BDE = \ (BDA + BAD). But the half 
sum of any two quantities being taken from the 
greater will leave the half difference: therefore 
ADE is the half difference of BDA and BAD. 

Now, (Rule 3, Art. 137,) BD or BC : BA : : rad. : tan. ADB ; 

and (demonstration to last rule) AB -f- BD : AB — BD : : tan. J (BDA -|- 
BAD) : tan. £ (BDA— BAD) : : tan. BDE : tan. ADE; but BDE being equal 
to 45°, its tangent == rad. 

And ADE = (ADB — 45°) . •. AB + BD : AB — BD : : r : tan. (ADB — 45°) ; 
but AB + BC : AB — BC : : tan. J (ACB -f- BAC) : tan. £ (ACB — BAC) ; 
whence r : tan. (ADB — 45°) : : tan. £ (ACB + BAC) : tan. £ (ACB — BAC). 

Examples. 

Ex. 1. In the course of a calculation I have found the 
logarithm of AB = 2.596387, that of BC = 2.846392: now, 
the angle B being 55° 49', required the side AC. 







Calculation. 






As AB 






A. 


C. 7.403613 


: BC 








2.846392 


: : Rad. 








10.000000 


: tan. x 




60° 38' 58" 




10.250005 


As rad. 






A. 


C. 0.000000 


: tan. (x - 


-45) 


15° 38' 58" 




9.447368 


: : tan. J (A + C) 


62° 5' 30" 




10.276004 


: tan. J (A • 


-C) 


27° 52' 28" 
A 89° 57 ; 58" 




9.723372 


Then, 










As sin. A 




- 89° 57' 58" 


A. 


C. 0.000000 


: sin. B 




55° 49' 




9.917634 


::BC 








2.846392 


: AC 




580.8 




2.764026 



Sec. IV.] NUMERICAL SOLUTION OF TRIANGLES. 73 

Ex. 2. Given the logarithms of BC and AC 3.964217 
and 3.729415 respectively, and the angle C = 63° 17' 24", to 
find AB. Ans. 8317. 

Ex. 3. Given the logarithms of AB and BC 1.963425 and 
2.416347, and the angle B = 129° 42', to find AC. 

Ans. 327.27. 

CASE 3. 

142. Given the three sides, to find the angles. 

Rule 1. 

Call the longest side the base, and on it let fall a perpendicular 
from the opposite angle. 

Then, as the base is to the sum of the other sides, so is the 
difference of those sides to the difference of the segments of the 
base. 

Half this difference added to half the base will give the greater 
segment, and subtracted will give the less segment 

Having the segments of the base, and the adjacent sides, 
the angles may be found by Rule 2, Art. 137. 

Demonstration. — Let ABC (Fig. 49) be the tri- Fig. 49. 

angle, AB being the longest side : with the centre 
C and a radius CB, the less of the other sides, 
describe a circle, cutting AB in E and AC in F 
and G. Draw CD perpendicular to AB. Then 
(3.3) DE = DB ; therefore AE is the difference 
of the segments of the base. 

Also, AG = AC + CB ; and AF == AC — CB. 

Now, (36.3. cor.,) AB . AE = AG . AF; 

whence (16.6) AB : AG : : AF : AE, 

or AB : AC + CB :: AC — CB: AD — DB. 

Examples. 

Ex. 1. Given the three sides of a triangle, — viz. : AB = 
467, AC = 413, and BC = 394, to find the angles. 




74 





PLANE TRIGONOMETRY. [Chap. IIL 1 


As AB 




467 


Ar. Co. 7.330683 J 


: AC + BC 


807 


2.906874 1 


::AC 


-BC 


19 


1.278754 I 


: AD 


-DB 


32.833 


1.516311 I 


J(AD- 


-DB) 


16.4165 


1 


JAB 




233.5 


1 


AD 




249.9165 


1 


BD 




217.0835 


1 


As AC 




413 


Ar. Co. 7.384050 1 


: AD 




249.9165 


2.397794 J 


: :r 






10.000000 J 


: cos. 


A 


52° 45' 44" 


9.781844 1 


As BC 




394 


Ar. Co. 7.404504 1 


: BD 




217.0835 


2.336627 | 


: : r 






10.000000 


: cos 


.B 


56° 33' 58" 


9.741131 



Whence C = 180 - (A+ B) = 70° 40' 18". 

Ex. 2. Given the three sides of a triangle, BC 167, AB 
214, and AC 195 yards, respectively, to find the angles. 

Ans. A = 47° 55 f 13", B = 60° 4' 19, C = 72° 0' 28". 

Ex. 3. Given AB = 51.67, AC = 43.95, and BC = 27.16, 
to find the angles. 

Ans. A = 31° 42' 42", B = 58° 16' 34", C = 90° 0' 44". 

Eule 2. ■ 

143. As the rectangle of two sides is to the rectangle of the 
half sum of the three sides and the excess thereof above the third 
side, so is the square of radius to the square of the cosine of half 
the angle contained by the first mentioned sides. 



Sec. IV.] 



NUMERICAL SOLUTION OF TRIANGLES. 



75 



Demonstration. — Let ABC (Fig. 50) Fig. 50. 

be a triangle, of which AB is greater 
than AC. Make AD = AC. Join DC, 
and bisect it by AEF. Draw EH paral- 
lel and equal to CB. Join HB, and pro- 
duce it to meet AEF in F. Then, since 
EH is equal and parallel to CB, BH is jjV 
equal and parallel to CE, (33.1.) 
Therefore F is a right angle. Again : 
since BH is equal to ED, and the angle 
EGD = BGH and EDG = GBH, (26.1,) DG = GB and EG = GH. 
describe a circle, and it will pass through F. 




On EH 



Now,2AK = 2AG4-2GK=AC+AD+2DG + 2GK=AC + AB-f-BC; 
or AK == J (AC -f AB + BC), 

and AI = AK— KI = J (AC + AB -f BC) — BC. 



But, (Rule 2, Art. 137,) As AD : AE 
and AB : AF 

whence (23.6) AB . AD : AE . AF 



: r : cos. DAE (cos. J BAC),' 
: r : cos. \ BAC ; 
: r» : cos. 3 \ BAC. 



But (36.3, Cor.) AE . AF = AK . AI = \ (AC + AB + BC) . \ (AC -f AB + 
BC) — BC; 



whence AB . AC : \ (AC + AB -f BC) • {\ (AC -f AB -f BC)— BC) 



cos. 



BAC. 



Examples. 

Ex. 1. Given AB == 467, AC = 413, and BC = 394, to find 
the angle C. 

Here, put s = half sum of the sides : we have s = 637 and 
5 — AB = 170; whence 



. n p ,fAC 413 
S AC - BC {bC 394 


A.C. 7.384050 


A.C. 7.404504 


. (s 637 
: 5 .( 5 -AB)j 5 _ AB170 


2.804139 
2.230449 


::E 2 


20.000000 


: cos. 2 JBCA 


2)19.823142 


J BCA = 35° 20' 9" 


9.911571 


BCA = 70° 40' 18". 





In the above calculation the R* and its logarithm might have been omitted, 
since we have to deduct 20 in consequence of having taken two arithmetical 
complements. The sum of the logarithms is divided by 2, to extract the square 
root, (Art. 16.) 



76 PLANE TRIGONOMETRY. [Chap. III. 

The rule may be expressed thus : — 

Add together the arithmetical complements of the logarithms 
of the two sides containing the required angle, the logarithm 
of the half sum of the three sides, and the logarithm of the excess 
of the half sum above the side opposite to the required angle : the 
half sum of these four logarithms will be the logarithmic cosine 
of half that angle. 

Ex. 2. Given AB = 167, AC = 214, and BC = 195, to find 
the angles. 

Ans. A = 60° 4' 22", B = 72° 0' 28", C = 47° 55' 16". 

Ex. 3. Given AB = 51.67, AC = 43.95, and BC = 27.16, to 
find the angles. 

Ans. A= 31° 42' 40", B = 58° 16' 28", C = 90° 0' 52". 



SECTION V. 
INSTRUMENTS AND FIELD OPERATIONS. 

144. The Chain. Gunter's Chain is the instrument 
most commonly employed for measuring distances on the 
ground. For surveying purposes, it is made 66 feet or 4 
perches long, and is formed of one hundred links, each of 
which is therefore .66 feet or 7.92 inches long. The links 
are generally connected by two or three elliptic rings, to 
make the chain more flexible. A swivel link should be 
inserted in the middle, that the chain may turn without 
twisting. In order to facilitate the counting of the links, 
every tenth link is marked by a piece of brass, having one, 
two, three, or four points, according to the number of tens, 
reckoned from the nearest end of the chain. Sometimes 
the number of links is stamped on the brass. The middle 
link is also indicated by a round piece of brass. 

The advantage of having a chain of this particular length 
is, that ten square chains make an acre. The calculations 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 77 

are therefore readily reduced to acres by simply shifting 
the decimal point. There being one hundred links to the 
chain, all measures are expressed decimally, which renders 
the calculations much more convenient. Eighty chains 
make one mile. 

In railroad surveying, a chain of one hundred feet long 
is preferred, the dimensions being thus at once given in 
feet. 

When the measurements are required to be made with 
great accuracy, rods of wood or metal, which have been 
made of precisely the length intended, are used. In the 
surveys of the American Coast Survey, the unit of length 
employed is the French metre, equal to the 10000000th part 
of the quadrant of the meridian. The metre is #9.37079 
inches = 3.280899 feet = 1.093633 yards long. 

It were much to be desired that the metre, or some other 
unit founded on the magnitude of the earth, or on some 
other natural length, such as that of a pendulum beating 
seconds at a given latitude, were universally adopted as the 
unit. The metre will probably gradually come into general 
use. 

To reduce chains and links to feet, express the links 
decimally and multiply by 66. Thus, 7 chains 57 links = 
7.57 chains are equal to 7.57 X 66 = 499.62 feet = 499 feet 
7.4 inches. 

To reduce feet and inches to chains, divide by 66, or by 6 
and 11. The inches must first be reduced to a decimal of a 

563.67 , 
foot. Thus, 563 feet 8 inches = 563.67 feet= eh. = 

8.54 chains. bb 

Instead of a chain of 66 feet, one of 33 feet, divided into 
fifty links, is sometimes used. This is really a half chain, 
and should be so recorded in the notes. The half chain is 
more convenient when the ground to be measured is 
uneven. 

145. The chain is liable to become incorrect by use; its 
connecting rings may be pulled open, and thus the chain 
become too long, or its links may be bent, which will 



78 PLANE TRIGONOMETRY. [Chap. III. 

shorten the chain. Every surveyor should, therefore, have 
a carefully measured standard with which to compare his 
chain frequently. According to the laws of Pennsylvania, 
such a standard is directed to be marked in every county 
town, and all surveyors are required to compare their chain 
therewith every year. 

If the chain is too long, it may be shortened by tighten- 
ing the rings ; if it is too short, which it can only become 
by some of the links having been bent or some rings 
tightened too much, these should be rectified. 

It has been found that a distance measured by a perfectly 
accurate chain is very generally recorded too long ; if then 
the chain is found slightly too long, say from one fourth 
to one third of an inch, it need not be altered, a distance 
measured with such a chain being more accurately recorded 
than if the chain were correct. 

In using the chain, care should be taken to stretch it 
always with the same force, or the different parts of the line 
will not be correctly recorded. Like all other instruments, 
it should be carefully handled, as it is liable to injury. 

146. The Pins. In using the chain, ten pins are necessary 
to set in the ground to mark the end of each chain measured. 
These are usually made of iron, and are about a foot or fif- 
teen inches long, the upper end being formed into a ring, 
and the lower sharpened that they may be readily thrust 
into the ground. Pieces of red and white cloth should be 
tied to the ring, to distinguish them when measuring through 
grass or among dead leaves. 

147. Chaining. This operation requires two persons. 
The leader starts with the ten pins in his left hand and 
the end of the chain in his right; the follower, remain- 
ing at the starting point and looking at the staff set up 
to mark the other end of the line, directs the leader to 
extend the chain precisely in the proper direction. The 
leader then sticks one pin perpendicularly into the ground 
at the end of the chain. They then go on until the follower 
comes to this pin, when he again puts the leader in line, 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 79 

who places a second pin. The follower then takes up the 
first pin, and the same operation is repeated until the leader 
has expended all his pins. When he has stuck his last 
pin, he calls to the follower, who comes forward, bringing 
the pins with him. The distance measured — viz. : ten 
chains — is then noted. The leader, taking all the pins, again 
starts, and the operation is repeated as before. When the 
leader has arrived at the end of the line, the number of pins 
in possession of the follower shows the number of chains 
since the last "out," and the number of links from the last 
pin to the end of the line, the number of odd links. Thus, 
supposing there were two "outs," and the follower has six 
pins, the end of the line being 27 links from the last pin, 
the length would be 26.27 chains. 

Some surveyors prefer eleven pins. One pin is then 
stuck at the beginning of the line, and at every "out" a 
pin is left in the ground by the leader. 

If the chain-men are both equally careful, they may 
change duties from time to time. If otherwise, the more 
intelligent and careful man should act as follower, that 
being much the more responsible position. 

148. Recording the " Outs." As every " out" indicates 
ten chains, — or fixe chains, if a two-pole chain is used, — it is 
of great importance to have them carefully kept. Various 
contrivances have been suggested for that purpose. Some 
chain-men carry a string, in which they tie a knot for every 
out; others place in one pocket a number of pebbles, and 
shift one to another pocket at each out. Either of these 
methods is sufficient if faithfully followed out. One rule, 
however, should be faithfully adhered to, — viz. : that the 
memory should never be trusted. The distractions to 
which the mind is subject in all such operations, necessarily 
call off the attention, so that a mere number, which has no 
associations to call it up, will be very likely to be forgotten. 

Perhaps the best method of preserving the "outs" is to 
have nine iron pins and five or six brass ones. The leader 
takes all the pins and goes on until he has exhausted his 
iron pins ; he then goes on one chain, and, sticking a 



80 PLANE TRIGONOMETRY. [Chap. III. 

brass pin, calls, " Out." The follower then advances, bring- 
ing the pins. He delivers to the leader the iron pins but 
retains the brass ones. On arriving at the end of the line, 
the brass pins in the follower's possession will show the 
number of "outs" and the iron pins the number of chains 
since the last "out." Thus, supposing he has six brass 
and eight iron pins, and that the end of the line is 63 links 
from the last pin, the distance is 68.63 chains. 

149. Horizontal Measurement. In all cases where the 
object is to determine the area or the position of points on 
a survey, the measurements must either be made horizon- 
tally, or, if made up or down a slope, the distance must be 
reduced according to the inclination. 

In chaining down a slope, the follower should hold his 
end of the chain firmly at the pin. The leader should then 
elevate his end until the chain is horizontal, and then mark 
the point directly under the end of the chain. This may be 
done by means of a staff four or five feet long, which should 
be held vertical, or by dropping a pin held in the hand with 
the ring downwards, or by a plumb-line. If the ground 
slopes much, the whole chain cannot be used at once. In 
such cases the leader should take the end of the half or the 
quarter, and, elevating it as before, drop his pin or make a 
mark. The follower then comes forward, and, holding the 
50th or 25th link, as the case may be, the leader goes for- 
ward to the end of another short portion of the chain, which 
he holds up, as before. A pin is left only at the end of 
every whole chain. 

Chaining up a slope is less accurate than chaining down, 
from the difficulty of holding the end still, under the strain 
to which the chain is subjected. The follower should always, 
in such cases, be provided with a staff four or five feet long, 
and a plumb-line to keep it vertical. If the slope is so steep 
that the whole chain cannot be used at once, the leader 
should take (as before) the end of a short portion, say one 
fourth, and proceed up hill. The follower then elevates his 
end, holding it firmly against the staff, which is kept vertical 
by the plumb-line. The leader, having made his mark, noti- 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 81 

fies the follower, who comes forward and holds up the 
same link that the leader used. He then goes forward as 
before. 

150. When great accuracy is required, the chaining should 
be made according to the slope of the ground, leaving stakes 
where there is any change of the slope, and recording the 
distances to these stakes in the note book. The inclination 
of the different parts being then taken, the horizontal dis- 
tance can be calculated. If a transit with a vertical arc is 
employed, the slope can be obtained at once, and the proper 
correction may be made at the time. The best way is to have 
a table prepared for all slopes likely to be met with, and 
apply the correction on the ground. Instead of deducting 
from the distance measured, it is best to increase the length 
on the slope, calling each length so increased a chain : the 
horizontal distance will then be correctly recorded. Thus, 
supposing the slope to be 5°, in order that the base may be 
1 chain the hypothenuse must be 1.0038 : the follower 
should therefore advance his end of the chain rather less 
than half a link. 

If a compass is used, it may be furnished with a tangent 
scale, to be described hereafter. 

The following table contains the ratio of the perpen- 
dicular to the base, the correction of the base for each 
chain on the slope, and the correction of the slope for each 
horizontal chain. If the corrections are made as the work 
proceeds, the last column should be used ; if in the field- 
notes after the work is done, the third column furnishes 
the data. 



82 



PLANE TRIGONOMETRY. 



[CHAP. III. 





Slope, 


Correction 


Correction 






Correction 


1 

Correction 


An S le - ner 


of base, in 


of hypoth. 


Angle. 


Slope. 


of base, in 


of hypoth. 


pei 




links. 


in links. 






links. 


in links. 


3° 1 


19.1 


—0.14 


+0.14 


17° 


1 : 3.3 


—4.37 


+4.57 


4° 1 


14.3 


0.24 


0.24 


18° 


1 


: 3.1 


4.89 


5.15 


5° 1 


11.4 


0.38 


0.38 


19° 


1 


2.9 


5.45 


5.76 


6° 1 


: 9.5 


0.55 


0.55 


20° 


1 


2.7 


6.03 


6.42 


7° 1 


: 8.1 


0.75 


0.75 


21° 


1 


2.6 


6.64 


7.11 


8° 1 


7.1 


0.97 


0.98 


22° 


1 


2.5 


7.28 


7.85 


9° 1 


6.3 


1.23 


1.25 


23° 


1 


2.4 


7.95 


8.64 


10° 1 


5.7 


1.52 


1.54 


24° 


1 


2.2 


8.65 


9.46 


11° 1 


5.1 


1.84 


1.87 


25° 


1 : 


2.1 


9.37 


10.34 


12° 1 


4.7 


2.19 


2.23 


26° 


1 . 


2.1 


10.12 


11.26 


13° 1 


4.3 


2.56 


2.63 


27° 


1 : 


2 


10.90 


12.23 


14° 1 


4.0 


2.97 


3.06 


28° 


1 : 


1.9 


11.71 


13.26 


15° 1 


3.7 


3.41 


3.53 


29° 


1 : 


1.8 


12.54 


14.34 


16° 1 


3.5 


• 3.87 


4.03 


30° 


1 : 


1.7 


13.40 


15.47 



151. Tape-Lines. A tape-line is sometimes used instead 
of a chain in measuring short distances. It is, however, 
very little to be depended on. If used at all, the kind that 
is made with a wire chain should be employed. It is 
much less liable to be stretched than those made wholly 
of linen. 

152. Chaining being one of the fundamental operations 
of surveying, whether for trigonometrical purposes or for 
the calculation of the contents, it has been described 
minutely. If correct measurements are needful, accurate 
notes are no less so. The chief points to be attended to in 
recording the measurements are precision and conciseness. 
Some of the most approved methods are given in Chap- 
ter IY. 

153. Angles. For surveying purposes horizontal angles 
alone are needed, since all the parts of the survey are re- 
duced to a horizontal plane ; but to fix the direction of a 
point in space not only the horizontal but vertical angles 
are required. With the aid of these, and the proper linear 
measures, its position may be fully determined. 



154. Horizontal angles are measured by having a plane, 
properly divided, and capable of being so adjusted as to be 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 83 

perfectly horizontal. Movable about the centre of this 
plane is another plane, or a movable arm, carrying a pair 
of sights or a telescope, which can be placed so that the 
line of sight may pass through the object. If then this 
line be directed to one object, and the position of the two 
plates or of the arm on the plate be noted by an index 
properly situated, and then be turned so as to point to 
another object, the angle through which the plate or the 
arm has turned will be the horizontal angle contained by 
two planes drawn from the centre of the instrument to the 
two objects. 

155. Vertical angles are measured by having a pair of 
sights or a telescope so adjusted as to move on a horizontal 
axis, the horizontal position of the sights or the telescope 
being indicated either by a plumb-line or a level. 



156. The transit with a vertical arc, or the theodolite, 
are so arranged as to perform both these offices. As a full 
understanding of the use of the different parts of these 
instruments is necessary to their proper management, we 
shall enter, considerably in detail, into a description of 
them. 

THE TRANSIT AND THE THEODOLITE * 

157. General Description. The Transit or the Theo- 
dolite (Figs. 51 and 52) consists of a circular plate, divided 
at its circumference into degrees and parts, and so sup- 
ported that it can be placed in a perfectly horizontal posi- 
tion. This divided circle is called the limb. An axis 
exactly perpendicular to this plate, bearing another cir- 
cular plate, passes through its centre. This plate is so 
adjusted as to move very nearly in contact with the former 
without touching it. By this arrangement the upper plate 
can be turned freely about their common centre. This 
plate carries a telescope Q, resting on two upright supports 
KK, upon which it is movable in a vertical plane. The 
telescope, having thus a horizontal and a vertical motion, 

* The author is indebted to Professor Gillespie's "Treatise on Land Sur- 
veying" for many of the features in his mode of presenting the subjects of the 
Transit and Theodolite, their verniers and their adjustments. 



84 



PLANE TRIGONOMETRY 



[Chap. III. 



THE TRANSIT. 

Fig. 51. 




Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 



85 



THE THEODOLITE. 

Fig. 52. 




86 PLANE TRIGONOMETRY. [Chap. Ill, 

can readily be pointed to any object. The second described 
plate has an index of some kind, moving in close proximity 
to the divided arc, so that the relative position of the plates 
may be determined. If then the telescope be directed to 
one object, and afterwards be turned to another, the index 
will travel over the arc which measures the horizontal angle 
between the objects. 

In order to place the plates in a perfectly horizontal posi- 
tion, levelling screws and levels are required: these, as 
well as the other parts of the instrument, will be fully 
described in their proper place. 

158. The above description applies to both instruments. 
The transit, however, is so arranged that the telescope can 
turn completely over; it can, therefore, be directed back- 
wards and forwards in the same line. If the same thing is 
to be done by the theodolite, the telescope must be taken 
from its supports and have its position. reversed. This ope- 
ration is troublesome, and is, besides, very apt to derange 
the position of the instrument. 

For surveying purposes, therefore, the transit is much to 
be preferred; and when the axis on which the telescope 
moves is provided with a vertical arc it serves all the pur- 
poses of a theodolite. 

The theodolite has a level attached to the telescope. This 
is not generally found in the transit. 

159. The accuracy of these instruments depends on several 
particulars : — 

1. By means of the telescope the object can be dis- 
tinctly seen at distances at which it would be invisible by 
the unassisted eye. 

2. The circle, with its vernier index, enables the observer 
to record the position of the telescope with the same degree 
of precision with which it can be pointed. 

3. There are arrangements for giving slow and regular 
motion to the parts, so as to place the telescope precisely in 
the position required. 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 87 

4. There are other arrangements for making the plates 
of the instrument truly horizontal. 

5. Imperfections in the relative position of the different 
parts of the instruments may be corrected by screws, the 
heads of some of which are shown in the drawings. 

However complicated the arrangements for performing 
these various operations may make the instruments appear, 
that complication disappears when they are viewed in detail 
and properly understood. 

160. In the figures of these instruments, V is the vernier, 
covered with a glass plate. In some theodolites the whole 
divided limb is seen. In others (and in the transit) but a 
small portion is exposed, — it being completely covered by 
the other plate, except the small portions near the vernier. 
Transits have generally but one vernier, though in some 
instruments there are two. The theodolite has generally 
two, and sometimes three or four. B is the compass box, 
containing the magnetic needle N". A, A, are the levels. 
C and D are screws ; the former of which is designed to 
clamp the lower plate, and the latter to clamp the plates 
together. T and U are tangent screws, to give slow and 
regular motion when the plates are clamped: by the 
former the whole instrument is turned on its axis, and by 
the latter the upper plate is moved over the other. P, P are 
the levelling plates; and S, S, S, are three of the four 
levelling screws. E is the vertical circle, with its vernier F. 
G is a level attached to the telescope. H is a screw to 
clamp the horizontal axis, (not visible in the figure of the 
theodolite,) and I a tangent screw, to give it regular 
motion. 

161. The Telescope. A telescope is a combination of 
lenses so adjusted in a tube as to give a distinct view of a 
distant object. It consists, essentially, of an object-glass, 
placed at the far end of the tube, and an eye-piece at the 
near end. 

By the principles of optics, the rays of light proceeding 
from the different points of the object are brought to a 



88 



PLANE TRIGONOMETRY. 



[Chap. III. 



focus within the tube, (Fig. 53,) there forming an 
inverted image. Crossing at this focus, they pro- 
ceed on to the eye-piece, by the lenses of which 
they are again refracted, and made to issue in 
parallel pencils, thus giving a distinct magnified 
image of the object. 

162. The Object-glass. Whenever a beam of 
light passes through a lens, it is not merely re- 
fracted, but it is likewise separated into the different 
colored rays of the solar spectrum. This separa- 
tion of the colored rays, or the chromatic aberration, 
causes the edges of all bodies viewed with such a 
glass to be fringed with prismatic colors, instead of 
being sharply defined. It has been found, how- 
ever, that the chromatic aberration may be nearly 
removed, by making a compound lens 
of flint and crown glass, as represented 
in Fig. 54, in which A is a concavo- 
convex lens of flint glass, and B a 
double convex lens of crown glass, — the 
convexity of one surface being made to 
agree with the concavity of the other 
lens. The two are pressed together by a screw 
in the rim of the brass box which contains them, 
thus forming a single compound lens. "When the 
surfaces are properly curved, this arrangement is 
nearly achromatic. 

The object-glass is placed in a short tube, 
movable by a pinion attached to the milled head 
W. (Figs. 51, 52.) By this means it may be moved 
backwards and forwards, so as to adjust it to dis- 
tinct vision. 



Fig. 53. 
D 



Fig. 54. 




163. The Eye-piece. The eye-piece used in 
the telescopes employed for surveying purposes 
consists of two plano-convex lenses, fixed in a 
short tube, the convex surfaces of the lenses being 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 89 

towards each, other. This arrangement is known as 
" Ramsden's Eye-piece." 

164. A telescope with an object-glass and an eye-piece 
as above described, inverts objects. By the addition of two 
more lenses the rays may be made to cross each other 
again, and thus to give a direct image of the object. As 
these additional lenses absorb a portion of the light passing 
through them, they diminish the brightness of the image. 
They may therefore be considered a defect in telescopes 
intended for the transit or theodolite. A little practice 
obviates the inconvenience arising from the inversion of 
the image. The surveyor soon learns to direct his assistant 
to the right when the image appears to the left of its 
proper position, and vice versa. 

165. The Spider-Lines. The advantage gained by the 
telescope in producing distinct vision, would add nothing to 
the precision of the observations, without some means of 
directing the attention to the precise point which should 
be observed in the field of view. The whole field forms a 
circle, in the centre of which the object should appear at 
the time its position is to be noted. This centre is de- 
termined by stretching across the field precisely in the focus 
of the eye-piece a couple of spider-lines or fine wires, at 
right angles to each other. The former are generally 
employed. When they are properly adjusted in the focus 
they can be distinctly seen, and the point to be observed 
can be brought exactly to coincide with their intersection. 
The magnifying power of the eye-piece enables this to be 
done with the greatest precision. When it has been 
effected, a line through the centre of the eye-piece and the 
centre of the object-glass will pass directly through the 
object. This line is called the line of collimation of the 
telescope. 

The spider-lines are attached by gum to the rim of a 
circular ring of brass placed in the tube of the telescope at 
the point indicated by the screw-heads a, a, (Figs. 51, 52,) 
some of which are invisible in the figure. These screws 



00 



PLANE TRIGONOMETRY. 



[Chap. III. 



serve to hold the ring in position, as 
represented in Fig 55, and to adjust 
it to its proper position. The eye- 
piece is made to slip in and out of the 
tube of the telescope, so that the focus 
may be brought to coincide exactly 
with the intersection of cross-wires. 
The perfect adjustment of the focus 
may be determined by moving the 
eye sideways. If this motion causes the wires to change 
their position on the object, the adjustment is not perfect: 
it must be made so before taking the observation. 




166. Spider-lines are generally used for making the 
"cross- wires," though platinum wires drawn out very fine 
are preferable. The wire is drawn to the requisite degree 
of fineness by stretching a platinum wire in the axis of a 
cylindrical mould and casting silver around it. The com- 
pound wire thus formed is then drawn out as fine as possi- 
ble and the silver removed by nitric acid. By this means 
Dr. Wollaston succeeded in obtaining wire not more than 
one thirty thousandth (gowo) of an inch in diameter. As 
such wire is very difficult to procure, the spider-threads are 
generally substituted. The operation of placing them in 
their proper position is thus performed. A piece of stout 
wire is bent into the form of the letter U, the distance between 
the legs being greater than the external diameter of the 
ring. A cobweb is selected having a spider hanging at the 
end. It is gradually wound round the wire, his weight 
keeping it stretched: a number of strands are thus obtained 
extending from leg to leg of the wire : these are fixed by a 
little gum. 

To iix them in their position, the wire is placed so that 
one of the lines is over notches previously made in the ring. 
The thread is then fixed in the position with gum or some 
other tenacious substance. The wire being removed, the 
line is left stretched across the opening in the proper 
position. 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 91 

167. The Supports. Attached to one of the horizontal 
plates, usually the index-plate of the instrument, are two 
supports, K, K, (Figs. 51, 52,) bearing the horizontal axis 
L. These supports should be made of precisely the same 
height, so that when the plate is level the axis may be hori- 
zontal. In some instruments there is an arrangement for 
raising or depressing one end of the axis so as to perfect 
the adjustment. In most cases, however, the adjustment 
is made perfect by the maker, and, if found not to be so, it 
must be remedied by removing the support which is too 
high and filing some of! from the bottom. This should 
always be done by the manufacturer. 

In the transit the telescope is attached immediately to the 
axis ; but in the theodolite the axis bears a bar M at right 
angles to it. This bar carries at its ends two supports, which 
from their shape are called Y' s > in "the crotch of which the 
telescope rests, being confined there by an arch of metal 
passing over the top. This arch is movable by a joint at 
one side, and is fastened by a pin at the other. By remov- 
ing the pin and lifting the arch the telescope is released and 
may be taken from the support. It rotates freely on its axis 
when confined by the arch. The telescope, being attached 
thus to the horizontal axis, admits of being elevated or 
depressed in a vertical plane so that it may be directed to 
any object. 

168. The Vertical Limb. In the theodolite, the vertical 
limb E consists of a semicircle of brass graduated on its 
face and attached to the bar M. This limb moves with the 
telescope upon the horizontal axis, and thus by means of 
the index F, serves to determine the angle of elevation of 
the object. In the transit with a vertical circle, the circle 
is attached to the end of the axis, as seen at E, the index 
then being attached to the support K. In some instru- 
ments, instead of the axis bearing a circle, an arc of from 
60° to 90° is attached to the support, and the index is fixed 
to the axis by an arm which is either permanently fastened 
to it or is capable of being clamped in any position. 



92 PLANE TRIGONOMETRY. [Chap. IIL 

169. The Levels. Attached to the horizontal plate are 
two levels A and A set at right angles to each other, so as 
to determine when that plate is horizontal. They consist 
of glass tubes very slightly curved, the convexity being 
upward. They are nearly rilled with alcohol, leaving a small 
bubble of air, which by the principles of hydrostatics will 
always take the highest point. If they are properly adjusted, 
the plate to which they are attached will, when these bub- 
bles have been brought to the middle of their run, be level, 
however it may be turned about its vertical axis. To the 
telescope of the theodolite and also to that of some transits 
another level Gr is fixed. This should be so adjusted that 
when the line of collimation of the telescope is horizontal 
the bubble may be in the centre of its run. 

170. The Levelling Plates. The four screws S, S, S, and 
S, called levelling screws, are arranged at intervals of 90° 
between the two plates P, P, which are called levelling 
plates or parallel plates. They screw into one plate and 
press on the other. By tightening one screw and loosening 
the opposite one at the same time, the upper plate, with the 
instrument above, may be tilted. To allow this motion, the 
column connecting them terminates in a ball, which works 
in a socket in the centre of the lower plate. A joint of this 
kind, called a ball-and-socket joint, allows movement in all 
directions. 

To level the instrument by means of these levelling 
screws, loosen the clamp, and turn the plates until the 
telescope is directly over one pair of the screws. Then, 
taking hold of two opposite screws, move them in contrary 
directions with an equal and uniform motion, until the 
bubble in the tube parallel to the line joining these screws 
is in the middle. Then turn the other screws in like man- 
ner until the other bubble comes to the middle of its tube. 
When they are both brought to this position the plates are 
level if the instrument is in adjustment. In levelling, care 
should be taken to move both screws equally. If one is 
moved faster than the other, the instrument will not be firm, 
or will be cramped. 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 93 

171. The Clamp and Tangent Screws. The former of 
these are used for binding parts of the instrument firmly 
together, the latter for giving a slow motion when they are 
so bound. The clamp C tightens the collar O clasping 
the vertical axis, and thus holds it and the plate attached 
to it firmly in their places. The other plate, moving on an 
axis within the former, may, notwithstanding, move freely. 
When this clamp is tightened, the collar may be moved 
slowly round by means of the tangent screw T. In its 
motion it carries with it the axis and attached plate. The 
clamp D fastens the two plates together. They may, how- 
ever, when so clamped, be made to move slightly on each 
other by means of the tangent screw TJ. If both clamps 
are tight, the instrument is firm, and the telescope can only 
be turned horizontally by one of the tangent screws. If 
the clamp C is tight and the other loose, the telescope and 
upper plate will move while the lower remains fixed. If D 
is tight and C is loose, the two plates are firmly attached 
to each other; but the whole instrument can be moved 
horizontally. 

Attached to the horizontal axis there is likewise a clamp 
H and tangent screw I, the purposes of which are similar 
to those described, — the clamp fixing the axis, and the 
screw moving it slowly and steadily. 

172. The "Watch-Telescope. Connected with the lower 
part of theodolites of the larger class there is a second tele- 
scope R, the object of which is to determine whether the in- 
strument has changed position during an observation. It 
is directed to some well defined object, and after all the ob- 
servations at the station have been made, or more frequently 
if thought necessary, it should be examined to see whether 
or not it has changed its position. If it has, the divided 
arc has changed also. The instrument, therefore, requires 
readjustment. 

173. Verniers. As it would be very difficult to divide a 
circle to the degree of minuteness to which it is desirable 
to read the angles, or, if it were so divided, since it would 



94 



PLANE TRIGONOMETRY. 



[Chap. III. 



be impossible for the eye to detect the divisions, some 

contrivance is necessary to avoid both difficulties. These 

difficulties will, perhaps, be made more striking by a 

simple calculation. The circumference of a circle 6 inches 

in diameter is 18.849 inches. If the circle is divided into 

360 

decrees there will be =19.1 divisions in the space 

6 18.849 * 

of an inch. If the divisions are quarter degrees there will 
be 76.4 to the inch ; and if minutes, there would be 1150 
divisions to every inch. The first and second could be 
read ; but the third, though it might by proper mechanical 
contrivances be made, yet it would be almost, if not en- 
tirely, impossible to distinguish the cuts so as to read the 
proper arc. And yet that division is not so minute as is 
sometimes desirable on a circle of that diameter. The 
vernier is a simple contrivance to effect this subdivision of 
space, in a way to be perfectly distinct and easily read. 

174. The principle of the vernier will be best understood 
by a simple example. In the adjoining figure, (Fig. 56,) 
AB represents a scale with the inch divided into tenths, the 
figure being on a scale of 3 to 2 or 1 J times the natural size. 

Fig. 56. 









/ S 














10 . o 






~i — r i 


D 


| 


















C 













TTT 

3 



















1 




2 


9 








1 










2 



CD is another scale having a space equal nine of the 
divisions on AB divided into ten equal parts. This second 
scale is the vernier. Kow, since ten spaces of the vernier are 
equal to nine of the scale, each of the former is equal to nine 
tenths of one of the latter. If then the on the vernier 
corresponds to one of the divisions of the scale, the first 
division of the vernier will fall ^ of a space or. ^ of an 
inch below the next mark on the scale, the next division 



Sec. V.] 



INSTRUMENTS AND FIELD OPERATIONS. 



95 



will fall ^ of an inch below, the next jjfo, and so on. The 
in the figure stands at 28.7 inches. 

If now the vernier be slid up so that the first division 
shall correspond to a division on the scale, the will have 
been raised ^ inch. If the second be made to coincide, 
the vernier will have been raised ^ of an inch. If it he 
placed as in Fig. 57, the reading will be 28.74 inches. 

Fig. 57. 





1 











5 


V 



























| 


























































3 

























2 


9 


















2|8 



The student should make for himself paper scales, di- 
vided variously, with verniers on other pieces of paper, so 
that he may become familiar with the manner of reading 
them. If his scale is to represent degrees, the portion re- 
presenting the arc might be drawn as a straight line, for the 
sake of facility in the drawing. It will illustrate the subject 
as well as if an arc of a circle were used. He should be- 
come particularly familiar with the one represented by Fig. 
60, as it is the division most commonly used in theodolites 
and transits. 



175. The Reading of the Vernier. To determine the 
reading of the vernier, — that is, the denomination of the parts 
into which it divides the spaces on the scale, — observe how 
many of the spaces on the scale are equal to a number on 
the vernier which is greater or less by one. The number of 
spaces on the vernier, so determined, divided into the value 
of one of the spaces on the scale, will give the denomination 
required. Thus, in Figs. 56 and 57, ten spaces of the ver- 
nier correspond with nine on the scale : the reading is 
therefore to ^ of ^ = ^ of an inch. 

If an arc were divided into half-degrees, and thirty spaces 
on the vernier were equal to twenty nine or to thirty one 



DQ PLANE TRIGONOMETRY. [Chap. III. 

spaces on the arc, the reading would be to ^ of \° = ^° = 1 
minute ; or, as it is usually expressed, to minutes. Fig. 60 
is an example of this division. 

176. To read any Vernier. First, determine as above 
the reading. Then examine the zero point of the vernier. 
If it coincides with any division of the scale as in Fig. 56, 
that division gives the true reading, — 28.7 inches. But if, 
as will generally be the case, it does not so coincide, note the 
division of the scale next preceding the place of the zero, and 
then look along the vernier until a division thereof is found 
which is in the same straight line as some division on the 
scale. This division of the vernier gives the number of 
parts to be added to the quantity first taken out. Thus, in 
Fig. 57, the of the vernier is between 8.7 and 8.8, and 
the fourth division on the vernier is in a line with a division 
on the scale : the true reading is therefore 28.74 inches. 

To assist the eye in determining the coincidence of the 
lines, a magnifying glass, or sometimes a compound micro- 
scope, is employed. 

When no line is found exactly to coincide, then there will 
be some which will appear equally distant on opposite sides. 
In such cases, take the middle one. 

177. Retrograde Verniers. Most verniers to modern 
instruments are made as above described. In some in- 
stances, the vernier is made to correspond to a number of 
spaces on the arc one greater than that into which it is 
divided. Such verniers require to be read backwards, and 
are hence called retrograde verniers. Fig. 58 is an ex- 
ample of one of this kind. It is the form that is generally 
used in barometers. It is drawn to one and a half times the 
natural size : the inches are divided into tenths, and eleven 
spaces on the scale correspond with ten on the vernier. 



Sec. V.] 



INSTRUMENTS AND FIELD OPERATIONS. 

Fier. 58. 



97 



/ 


















1 












, 

















1 I 1 I 






















































3 

























2 


9 














2 


8 



The value of one division of the vernier is y 1 ^ inch. If 
therefore on the vernier, corresponds to a division on the 
scale, 1 on the vernier will be ^ of an inch below the next 
on the scale, 2 will be ^ below; and so on. If the vernier 
is raised so that the 1 on the vernier is in line, it is raised 
i^o i n ch ; if 2 is in line, it is raised ^ ; and so on. The 
reading in Fig. 58 is 29.7 inches, and in Fig. 59, 29.53 
inches. 

Fig. 59. 












V 




I 6 












1 



















1 




1 














| 






1 




























3 



















2 


9 


















2 


8 





178. In Fig. 60, the arc is divided by the longer lines into 
degrees, and by the shorter into half degrees, or 30' spaces. 



Fig. 60. 




98 



PLANE TRIGONOMETRY. 



[Chap. III. 



Thirty spaces on the vernier are equal to twenty nine on 
the arc. The reading is therefore to -^ of 30 minutes = 1 
minute. The zero of the vernier stands between 41° 30' 
and 42°. On looking along the vernier, it is seen that the 
fifth and sixth lines coincide about equally well. The ver- 
nier therefore reads 41° 35 r 30" 



179. Reading backwards. Sometimes it is required to 
read backwards from the zero point on the limb. When 
this is done, the numbers on the vernier must be read in 
reverse, the highest being called zero, and the zero the 
highest. 

Fig. 61. 





Thus, in Fig. 61, the zero of the vernier standing to the 
right of 360 on the limb, between 1° 30' and 2°, and the 
division marked with an arrow-head being in line, the angle 
is 1° 41/. This mode of reading is needful when using the 
theodolite to take angles of depression, and also when using 
the transit to trace a line that bends backwards and for- 
wards, the angle of deflection being then generally taken, 
and recorded to the right or to the left, as the case may be. 



180. Double Verniers. To avoid the inconvenience of 
reading backwards, a double vernier is frequently made. It 
consists of two direct verniers having the same zero point, 
as shown in Fig. 62. 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 99 

Fig. 62. 




The arc in this figure is divided into degrees, and eleven 
spaces on the arc are equal to twelve on the vernier : the 
reading is therefore to 5 minutes. When the figures on the 
arc increase to the right, the right-hand vernier is used, and 
vice versa. The reading on the figure is 2° 45' to the left. 

181. Another form of double vernier is shown in Fig. 63. 

Fig. 63. 




In the figure, the vernier reads to minutes. When the 
zero of the vernier is to the left of that on the limb, the 
figures begin at the zero and increase towards the left to 
15' ; they then pass to the right-hand extremity, and again 
proceed to the left ; that is, they stop at A and commence 
again at B. The upper figures of each half are the con- 
tinuation of the lower figures of the other half. The read- 
ing in Fig. 63 is 1° 8' to the left, 

In Fig. 64 the reading is 3° 19' to the right. 



100 



PLANE TRIGONOMETRY. 

Fig. 64. 



[Chap. III. 




Fig. 65. 
E 



182. If the preceding descriptions have been thoroughly 
understood, the student will have no difficulty in reading 
the arc on any limb, however it may be divided. He should 
study the different positions until he can determine the 
angle with readiness, however the index may be placed. 
For this purpose, as before remarked, he should make for 
himself verniers with different scales, so that they can be 
placed in various positions. 

The construction of such verniers is very simple. Suppose, 
for example, it is desired to divide the arc into degrees and 
subdivide it by the vernier so as to read to 5 minutes : twelve 
spaces on the vernier must equal eleven on the arc, or one 
space on the vernier will equal ^ of a space on the arc. Let 
(Fig. 65) E be the centre and AB a por- 
tion of the limb, which, for the purpose 
intended, should not be of less radius 
than ten or twelve inches, and let CD be 
the vernier; with some other radius EG-, 
which should be greater than EB, de- 
scribe an arc GF; take EI : EG : : number A 
of divisions on the vernier : the number f 
that occupies the same space on the arc, H 
— in this case, as 12 to 11. Take from 
the table of chords the chord of 1° or ^°, as the case may be, 
and multiply it by the length of EG ; lay off the product on 
GF, thus determining the points 1, 2, 3, &c, and lay off the 
same length on IH, determining the points a, b, c, &c. ; stick 
a fine needle in the centre E; then, resting the ruler against 
the needle, bring it so as to coincide with I, and draw the 




Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 101 

division on AB ; then, keeping it pressed against the needle, 
bring it successively to the other points on GIF, and draw 
the corresponding divisions on AB. The arc will then be 
divided. In the same way, resting the ruler against the 
needle, and bringing it successively to the points on IH, the 
vernier may be divided. The reason of this process is, that 
since oh = 1.2, the degrees of ah will be to the degrees of 
1.2 as the radius of GF is to the radius of HI, as 11 to 12. 
Hence each division of the vernier is ^ of one division of 
the arc. 

By this means the divisions may be made with facility 
and accuracy. 

183. Adjustments. In order that the theodolite and 
transit may give correct results when used, it is necessary, 
that the different parts should bear the precise relations to 
each other that they are intended to have. By the term 
adjustment is meant the due relation of the parts to each 
other : when it is said an instrument is in adjustment, it is 
meant that every part bears to every other precisely its 
proper relations, so that the instrument is in perfect work- 
ing order. 

Before making any observations with a new instrument, 
it should be carefully examined to verify the adjustment. 
If the parts are not found to be properly adjusted, they 
must be rectified. 

184. For measuring horizontal angles, the following con- 
ditions are necessary :— 

1. The levels should be parallel to the plates, so that 
when the bubbles are in the middle of their run, the plates 
shall be horizontal. 

2. The axes of the two horizontal plates should be per- 
fectly parallel and perpendicular to the plane of the plates. 

3. The line of collimation should be perpendicular to the 
horizontal axis. 

4. The horizontal axis should be parallel to the plane of 
the plates, so that when they are horizontal it may be so 
likewise. 



102 PLANE TRIGONOMETRY. [Chap. III. 

185. First Adjustment. The levels should be parallel to 
the horizontal plates. 

Verification. Clamp the two plates together ; loosen the 
clamp C, (Figs. 51, 52 ;) bring the telescope directly over one 
pair of levelling screws, and level the plates as directed in 
Art. 170. Turn the plates half round : if the bubbles retain 
their position, the plane of the levels is perpendicular to 
the axis on which the lower plate turns. If either of them 
inclines to one end of its tube, it is out of adjustment, and 
requires rectification. 

To rectify the fault, bring the bubble halfway back to the 
middle by means of the capstan screw attached to one end, 
and the other half by the levelling screws. Again reverse 
the position of the plate : if the bubble now remains in the 
middle, the rectification is complete ; if not, the operation 
must be repeated. When both levels have been so arranged 
that the bubbles retain their position in the middle of their 
run when the plates are turned all round, the adjustment 
is perfect, and the axis is perpendicular to the plane of the 
levels. 



186. Second Adjustment. The axes of the horizontal 

plates should be parallel. 

Verification. Level the plates, as directed in last article. 
Clamp the lower plate, and loosen the vernier-plate. Turn 
it half round : if both bubbles still retain their position the 
axes are parallel. If the plates move freely over each 
other without binding in any position, they are perpendi- 
cular to the axes, or, at least, the upper one is so. 

If any defects be found in either of these particulars, 
the instrument should be returned to the maker to be 
rectified. 



187. Third Adjustment. The line of collimation of the 
telescope of the theodolite should be parallel to the common axis 
of the cylinders on which it rests in its Y '$• 



Sec. V.] 



INSTRUMENTS AND FIELD OPERATIONS. 



103 



Verification. Direct the telescope so that the intersec- 
tion of the wires bisects some well defined point at a dis- 
tance. Rotate the telescope so as to bring the level to the 
top. If the intersection still coincides with the object, 
the adjustment is perfect. If it has changed its posi- 
tion, bring it half-way back, by the screws a, a, and verify 



again. 




188. Fourth Adjustment. The line of collimation must be 
perpendicular to the horizontal axis. 

Verification for the Transit Set the transit on a piece 
of level ground, as at A, (Fig. 66,) and level it carefully. 
At some distance — say four or five chains — set a stake B 
in the ground, with a nail driven in the head, and direct 
the telescope so that the cross- Fi s- 66 « 

wires may bisect exactly on the 
nail. Clamp the plates, turn 
the telescope over, and place a 
second stake C precisely in the 

line of sight. If the adjustment is perfect, the three points 
B, A, and C will be in a straight line. To determine 
whether they are so, turn the plate round until the tele- 
scope points to B ; turn it over, and, if the line of sight 
passes again through C, the adjustment is perfect. If it 
does not, set up a stake at E, in the line of sight : then the 
prolongation of the line BA bisects EAC. 

Let FG (Fig. 67) be the 
horizontal axis. Then, if 
the line of collimation 
makes the angle FAB c 
acute, when the telescope 
is turned over it will make 
FAC = FAB. The angle 
CAD is therefore equal to 
twice the error. Now, if the plate is turned until the line 
of sight is directed to B, the axis will be in the position 
F'G'. Turn the telescope over, and the angle EAF'= 
F'AB ; CAE is therefore equal to four times the error. 



D- 




G F 



104 PLANE TRIGONOMETRY. [Chap. III. 

Hence, to rectify the error, the instrument being in the 
second position, place a stake at H, one fourth of the dis- 
tance from E to C, (Fig. 67,) and, by means of the screws 
a, a, (Fig. 51,) move the diaphragm horizontally till the 
vertical line passes through II. Verify the adjustment; 
and, if not precisely correct, repeat the operation. 



189. The above method is inapplicable to the theodolite, 
as its telescope does not turn over. For the means of 
detecting and correcting the error, see Art. 190. 



190. Fifth Adjustment. The horizontal and the vertical 
axes should be perpendicular. 

Verification for the Transit. Suspend a long plumb-line 
from some elevated point, allowing the plummet to swing 
in a bucket of water ; then level carefully, and bisect the 
line accurately by the vertical wire. If, on elevating and 
depressing the telescope, the line is still bisected, the ad- 
justment is good. If not, the error may be corrected by 
filing one of the frames. Instead of a plumb-line, any ele- 
vated object and its image, as seen reflected from the surface 
of mercury or of molasses boiled to free it from bubbles, 
may be employed. 

Verification for the Iheodolite, If the instrument, treated 
as above, shows a defect, the error may be either in the 
axis, or in the position of the Y's. To determine which, 
turn the plates half round, and reverse the telescope. If 
the deviation is now on the same side as before, the Y's are 
in fault. Their position in most instruments may be cor- 
rected by screws which move one of them laterally. If the 
line deviates to the opposite side from before, the position 
of the axis may be corrected by filing, as directed for the 
transit. 

This adjustment may also be examined by directing the 
telescope to some well defined elevated object, and then to 



Sec. V.] 



INSTRUMENTS AND FIELD OPERATIONS. 



105 



another on or near the ground. If none such can be 
found, let one be placed by an assistant ; then reverse the 
telescope in its Y's if the instrument is a theodolite, or 
turn it over if the instrument is a transit, and direct it to 
the upper object. If the cross-wires still intersect upon 
the lower point when the tube is depressed, the adjustment 
is perfect. 

191. Adjustments of the Vertical Limb. Having 
verified the various adjustments for horizontal motion, as 
described in the preceding articles, and rectified them if 
defective, the instrument is ready for use for horizontal 
work. To take angles of elevation, or to use the instru- 
ment for levelling, the following adjustments must also be 
examined : — 

1. The level beneath the telescope must be parallel to the 
line of collimation. 

2. The zero of the vernier must coincide with the zero 
of the vertical limb when the plates are level and the tele- 
scope horizontal. 



192. First Adjustment. The level must be parallel to the 
line of collimation. 

Verification. Select a piece of level ground, and drive 
two stakes, A and B, (Fig. 68,) four or &ve chains apart. 
At C, equidistant from them, set the instrument. Level 
the plates, and bring the bubble in the telescope level, to 
the middle of its run ; then let an assistant hold a graduated 
staff on A. Note exactly the point in which the line of 
sight meets the staff: then let the assistant remove the 
staff to B, and drive the stake B until the telescope points 



Fisr. 68. 




to the same spot on the staff. The tops of A and B are 
then level, whether the instrument is in adjustment or not. 



106 PLANE TRIGONOMETRY. [Chap. III. 

!Now remove the instrument to G, and level as before. 
Direct the telescope to the staff on B, and note the point 
I of intersection. Let the assistant carry the staff to A. 
Again note the intersection K. If the instrument is 
properly adjusted, these two points will coincide. If 
they do not, the line of collimation points too high or too low. 

Take the difference between BI and AK: This differ- 
ence will be LK, the difference of level as given by the 
instrument at G. Then say, As the distance between the 
stakes (BA) is to the distance from the instrument to the 
far stake (GA), so is the difference of apparent level of the 
stakes (LK) to the correction on the far staff (MK). 

This correction — either taken from the height AK if too 
great, or added to it if too small — will give AM, the height 
of a point on the same level as the instrument. Direct the 
telescope to this point, and rectify the level, by raising or 
lowering one end by means of the capstan screw until the 
bubble is in the middle of its run. If the operation has 
been carefully done, the adjustment is perfect. Verify 
again ; and, if needful, repeat the operation. 

193. Second Adjustment. The zeros of the vernier and 
of the vertical limb should coincide when the telescope is level. 

When the first adjustment is perfected, and the telescope 
is still level, examine the reading on the vertical limb care- 
fully: if the zeros coincide, the vernier is properly ad- 
justed ; if they do not, note the error, and have it marked 
somewhere on the instrument under the plates, that it may 
not be forgotten. It must be applied to all angles of eleva- 
tion taken by the instrument. 

If the index-arm is movable, as is frequently the case 
with transits, it should be adjusted before taking vertical 
angles. 

194. When all the preceding adjustments have been exa- 
mined, and rectified if necessary, the instrument is ready 
for work. It would be well, however, to examine carefully 
the reading of the verniers, to see that they are properly 
divided. However placed, no two lines of the vernier 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 107 

except the first and last should coincide with divisions on 
the arc. If two are found to do so in any position, 
there is an imperfection in the graduation. If the division 
is very fine, a number of lines in the immediate neighbor- 
hood of the coincident lines will differ very slightly from 
coincidence; but, when carefully examined with a good 
magnifier, they should recede gradually. 

Place the instrument where a good view of a fine point, 
some eight or ten chains distant, can be obtained. Level 
carefully, direct the line of sight to the point, and note the 
reading on the horizontal limb. Reverse the telescope in 
its Y 's, or, if the instrument is a transit, turn it over ; turn 
the vernier-plate till the line of sight passes again through 
the point, and note the reading. It should differ by 180° 
from that before obtained. If it does not, the divisions are 
not perfect, or the telescope is not over the centre of the 
plates. Either defect should condemn the instrument, as it 
can be remedied only by the maker. This verification 
should be tried in various positions of the divided plate. 
If these tests, and those formerly mentioned, are found to 
detect no imperfection, the instrument may be pronounced 
a good one. 

195. Taking Angles. Set the instrument precisely over 
the angular point, and level it, being careful to have the 
levelling screws pressed tightly against the plates, that the 
instrument may be steady. Set the index to zero, and 
clamp the plates, and, if there be more than one vernier, 
note the minutes and seconds of the others. Loosen the 
lower clamp, and bring the telescope so that the wires may 
intersect on the left-hand object; clamp, and perfect the ad- 
justment by the tangent screw. If there is a watch-tele- 
scope, set it upon some well-defined object, — such as a light- 
ning-rod or the corner of a chimney, — and clamp it tightly. 
Loosen the vernier-plate, and turn the telescope to the 
other object, perfecting the adjustment by the tangent 
screw. Examine the watch telescope, and, if the instru- 
ment has shifted, bring it back by the tangent screw, 
and readjust the telescope by moving the vernier-plate. 



108 



PLANE TRIGONOMETRY. 



[Chap. III. 



Now read the arc by the same index as before, noting the 
minutes and seconds by the other verniers. Take the mean 
of the minutes and seconds of each position for the true 
reading. Then the true reading in the first position taken 
from that in the second will give the angle required. It is 
convenient to have a table prepared, with the requisite 
number of columns, in which to set down the readings of 
the different verniers. Thus, suppose there were three 
verniers, 120 degrees apart : rule a table, with six columns, 
as below : — 



Occd. 

Sta. 



A 
A 



Obs. 

Sta. 



B 

C 



A 


B 


c 


0° 0' 0" 
75° 8' 15" 


0' 30" 

8' 0" 


59' 45" 
8' 30" 



Mean. 



0° 0' 7|" 

75° 8' 15" 



The first column is the occupied station; the second, the 
observed station ; the next three the readings of the verniers, 
and the sixth the mean. 

In the case above, the angle BAC would be 75° 8 r 7J". 
The instrument is supposed to read to 30", the 15" being 
taken when two lines on the vernier appear equally near 
coincidence. 



196. Repetition of Angles. The following method of 
observation is sometimes employed. Suppose the angle 
ABC is to be measured, A being the left-hand object : direct 
to A, and turn to B as above directed. Clamp the vernier- 
plate and loosen below, and bring the telescope again to A. 
Clamp below, loosen the vernier, and bring the telescope 
again to B. The index has now traversed an arc measuring 
twice ABC. The operation may be repeated as often as 
desired, noting the number of whole revolutions the tele- 
scope has made. Then divide the whole number of degrees 
by the number of repetitions. The result will be the 
degrees of the angle required. If there is a watch-telescope, 
it should be set carefully before each observation. When 
this is done, and proper care is taken to avoid deranging 



Sec. V.] INSTRUMENTS AND FIELD OPERATIONS. 109 

the instrument, the result may be depended on as more 
accurate than any single reading. Any error in the final 
reading, being divided by the number of observations, will 
affect the result by but a small part of its value. 

197. Verification of the Angles. When it is possible 
to do so, all the angles of a triangle should be measured. 
If their sum does not make 180°, there must be an error 
somewhere. Should the error be considerable, the work 
ought to be reviewed. But if it does not exceed two or 
three minutes, providing the instrument only reads to 
minutes, it may be distributed equally among the three 
angles, should there be no reason to suppose one is more 
accurate than another. But if more observations have been 
taken for some angles than for others, their determination 
should be most depended on, and a proportionally less part 
of the correction assigned to them. Suppose, for example, 
the angle A is the mean of five observations, B of three, 
while at C but one was taken, the error being 1/ 45" : we 
would proceed thus : — As J + J + 1 : f : : V 45" : 14", the 
correction for A. In the same manner the correction for B 
would be found to be 23", and for C, V 08". 

198. Reduction to the Centre. Where the objeet that 
has been observed is a spire or other portion of a building, 
it is impossible to set the instrument underneath the signal. 
In such cases, the observed angle must be reduced to what 
it would have been had the station been at the proper point. 
Thus, let C (Fig. 69) be the correct Fig. 69. 
station, and D the occupied station, 
which should be taken as near as 
possible to C. Take the angle ADB. 
Then if A, C, D, and B are all in the 
circumference of a circle, this will be 
equal to ACB. The station should 
be assumed as near this as possible. Calculate BC and AC 
from the distance AB and the angles observed at A and B. 
Also measure DC, either directly or by trigonometrical 
methods to be explained hereafter, and take ADC. 




110 PLANE TRIGONOMETRY. [Chap. III. 

Then, (Art. 139,) As CA : CD : : sin. ADC : sin. CAD. 

And as CB : CD : : sin. BDC : sin. CBD. 

Hence, ACB - AEB — CAD = ADB + CBD — CAD, 
becomes known. 

Example. Let CA = 9647 ft. ; CB = 8945 ft. ; ADB = 
68° 45'; DC = 150 ft,; and ADC = 97° 37'. 

As CA 9647 ft. A. C. 6.015608 

: CD 150 ft. 2.176091 

: : sin. ADC 97° 37' 9.996151 



: sin. CAD 


0° 52' 59" 


8.187850 


Ls CB 


8945 ft. 


A. C. 6.048420 


: CD 


150 ft. 


2.176091 


: sin. CDB 


166° 22' 


9.372373 


: sin. CBD 


0° 13' 35" 


7.596884 



Whence ACB = ADB + CBD — CAD = 68° 5' 36". 

199, Angles of Elevation. In measuring angles of ele- 
vation, the instrument must first be levelled; the telescope 
being then directed to the object, the reading of the vernier 
corrected for the index-error will be the angle of elevation. 



SECTION VI. 



MISCELLANEOUS PROBLEMS TO ILLUSTRATE THE RULES 
OF PLANE TRIGONOMETRY. 

Problem 1. Being desirous of determining the height 
of a fir-tree standing in my garden, I measured 100 feet 
from its base, the ground being level. I then took the 
angle of elevation of the top, and found it to be 47° 50' 30". 
Required the height, the theodolite being 5 feet from the 
ground. 



Sec. VL] 



MISCELLANEOUS PROBLEMS. 



Ill 



Fig. 70. 




Solution. 

Make AB (Fig. 70) equal to 100 feet; 
draw AD and BC perpendicular to AB, 
making the former five feet from the same 
scale. Draw DE parallel to AB, and 
make EDO = 47° 50', the given angle. 
Then will CB be the height of the tree. 

Calculation. 

As rad. : tan. EDO : : DE : EC = 110.45 feet 
whence BC = 110.45 + 5 = 115.45. 

Problem 2. One corner C (Fig. 71) of 
a tract of land being inaccessible, to de- 
termine the distances from the adjacent 
corners A and B, I measured AB = 9.57 
chains. At A, the angle BAC was 52° 19' 
15", and at B, the angle ABC was 63° 19' 
45". Eequired the distances AC and BC. B 

Calculation. 

As sin. ACB (64°21') : sin. A (52° 19' 15") : : AB (957) : 
BC = 840.2 links. As sin. ACB (64° 21') : sin. B (63° 19' 
45") : : AB : AC = 948.7 links. 





Problem 3. In measuring the sides Fi s- 72 - 

of a tract of land, one side AB (Fig. 
72) was found to pass through a swamp, 
so that it could not be chained. I there- 
fore selected two stations, C and D, on 
fast land, and took the distances and angles as follows, — 
viz.: AC = 37.56 chains; CD = 50.25 chains; BAC = 
65° 27' 30"; ACD = 123° 46' 20"; CDB = 107° 29' 15": 
the corner B being inaccessible, the distance BD could not 
be measured. Required AB. The angle CDA could not be 
taken, owing to obstructions. 



112 



PLANE TRIGONOMETRY. 



[Chap. III. 



Solution. 

Join AD. Then, from the triangle ACD. we have, (Art. 

140,) 

CAD 4- CDA 
As CD + CA (87.81) : CD - CA (12.69) : : tan. 1 



(28° 6' 50") : tan. 



CAD - CDA 



= 4° 24' 54"; 



whence CAD = 28° 6' 50" + 4° 24' 54" = 32° 31' 44", 
and CDA = 28° 6' 50" - 4° 24' 54" = 23° 41' 5G" ; 

then, sin. CDA : sin. ACD : : AC : AD = 77.68. 

Now, in ADB we have AD = 77.68, the angle DAB = CAB 
__ CAD = 32° 5b' 46", and the angle ADB = BDC - ADC 
= 83° 47' 19", to find AB; thus, 

As sin. B : sin. ADB :: AD : AB = 86.455 chains. 



Fig. 73. 



Problem 4. To determine the position of a point D on 
an island, I ascertained the distances of three objects on the 
main land as follows:— AB = 248.75 chains, BC = 213.25 
chains, and AC = 325.96 chains. At D the angle ADB was 
found to be 29° 15', and BDC 20° 29' 30". Eequired the 
distance of D from each of the objects. 

Construction. 

With the given distances construct 
the triangle ABC. At C and A make 
the angles ACE = 29° 15', and CAE 
= 20° 29' 30". About AEC describe 
the circle ACD. Join EB, and pro- 
duce it to D, which will be the point 
required. 

For (21.3) ADB = ACE = 29° 15', 
and CDB = CAE = 20° 29' 30". 

Calculation. 

1. In ABC we have the three sides to find the angle BAC 
= 40° 51' 30". 

2. In CAE we have the angles and side AC to find the 
side AE = 208.705. 

3. In BAE we have BA, AE, and the included angle 
BAE, to find ABE = 50° 55 f 48", AEB = 67° 43' 12". 




Sec. VI.] 



MISCELLANEOUS PROBLEMS. 



4. In ABD we have the angles and side AB, to find AD 
= 395.24 and BD = 188.07. 

5. In A CD we have the angles and sides AC, to find CD 
= 379. 




Problem 5.— Wishing to obtain the distance between tw 
trees, C and D, situated on Fig. 74. 

the side of a bill, and not 
being able to find level 
ground for a base, I select- 
ed a gradual slope, on which 
I measured the distance AB 
(Fig. 74) 400 yards. I then 
took the horizontal and ver- 
tical angles as follow: — At 
A, the angle BAD was 101° 
47' 15", BAC 39° 25' 45". The elevation of B was 5° 32' 
45", of C, 8° 19' 30", and of D, 12° 29'. At B, the 
angle ABD was 59° 13' 15", and ABC 125° 36' 45". 

Required the distance CD, and the elevations of C and 
D above A. 

Conceive a horizontal plane to pass through A, meeting 
vertical lines through B, C, and D in the points E, F, and G. 
Then, since the angular distances are measured horizontally, 
we have the following angles given, — viz. : EAG = 101° 47' 
15", EAF = 39° 25' 45", AEG = 59° 13' 15", and AEF = 
125° 36' 45". 

Calculation. 

1. To find AE, we have r : cos. BAE (5° 32' 45") : : AB 
(400) : AE = 398.13. 

2. To find AG. As sin. AGE : sin. AEG : : AE : AG = 
1051.07, log. 3.021631. 

3. To find AF. As sin. AFE : sin. AEF : : AE : AF - 
1253.96, log. 3.098284. 

4. TofindFG,(Art.l41.)AsAG:AF::r:tan.z==50°l'49". 
And, as rad. : tan. (x — 45°) : : tan. J (AGF + AFG) : tan. 

J (AGF - AFG) = 8° 16' 34"; 

then AGF = 58° 49' 15" + 8° 16' 34" = 67° 5' 49", 

and AFG = 58° 49' 15" - 8° 16' 34" = 50° 32' 41". 

8 



114 PLANE TRIGONOMETRY. [Chap. III. 

Then, as sin. AGF : sin. FAG : : AF : GF = 1205.9. 

5. To find GD and CF. As r : tan. GAD : : AG : GD - 
232.69 = Elevation of D. 

And as r : tan. CAF : : AF : FC = 183.49 = Elevation of C. 

6. To find CD. CD = y/ CH 2 + HD 3 = 1206.9 = Dis- 
tance of CD. 

Problem 6. — Being desirous to determine the height of 
a tower standing on the summit of a hill, I measured 75 
yards from its base down the declivity, which was a regular 
slope. I then took the elevation of the top, 49° 37' 45", and 
of the bottom, 8° 19', the height of the instrument being 5 
feet. What was the height of the tower ? Ans. 76.44 yds. 

Problem 7. — To determine the height of a tree in an 
inaccessible situation, I took a station, and found the ele- 
vation of the top to be 38° 45' 15" ; then, measuring back 
100 feet, the elevation was found to be 24° 18'. Required 
the altitude of the tree and its distance from the first sta- 
tion, the instrument being 4 feet 9 inches high. 

Ans. Height, 107.95 feet; distance, 128.57 feet. 

Problem 8. — To determine the distance of two objects 
A and B, I took two stations C and D, distant 35.75 chains, 
from which both could be seen. At C, the angle ACD was 
found to be 103° 47', and BCD 45° 29' 30" ; at D, the angle 
BDC was 110° 23' 30", and ADC 60° 21' 15". Required the 
distance AB. Ans. 99.236 ch. 

Problem 9.— The side AB (Fig. 75) of a tract of land 
being inaccessible, and not being able to find two stations 
from which both ends were visible, 

Fig. 75. 

I measured two lines, CD, 7.75 ch. 



"> 



and DE, 7.92 ch., and took the angles 
as follow : At C, the angle ACD was 
68° 15'. At D, CDA was 50° 27', 
ADB 112° 46', and BDE 43° 30'. At 
E, DEB was 75° 10'. What was the length of AB ? 

Ans. 14.10 ch. 




Sec. VI.] 



MISCELLANEOUS PKOBLEMS. 



115 



Problem 10. — To determine the position of a point D, 
situated on an island, I took the angles to three objects, 
A, B, and C, situated on the shore, and found them to be 
ADB, 19° 14' 30", CDB, 24° 19'. I subsequently deter- 
mined the distances AB = 4596 yards, AC = 5916 yards, 
and BC = 4153 yards. Required the distance of D from 
each of the objects, it being nearest to B. 

Ans. AD = 828T.2 yards ; BD = 4127.7 yards ; CD = 
7550.8 yards. 

Problem 11. — To determine the height of a mountain 
rising abruptly from the water of a lake, I selected a station 
C on the slope of the hill rising from the opposite shore, and 
took the angle of elevation of the summit, 47° 22' 15", and 
depression of the water's edge at the base of the mountain 
in the vertical plane through the summit, 12° 30'. Then 
measuring up the slope, directly from the rock, a distance of 
800 yards, to a station D, the elevation of the summit was 
25° 33' 30", the depression of the water's edge, 18° 15', and 
of the top of a staff left at C to mark the height of the 
instrument, 24° 15'. Required the height of the mountain. 

Ans. Height, 1390.7 yds. 



Fig. 76. 



Problem 12. — To determine the heights and distance of 
two trees C and D, standing on a hill side, I measured on level 
ground a base line AB 252.28 feet 
long, and took the following angles : 
At A, the angle of position of C from 
B was = 82° 54' 30", and of D from 
B = 89° 24'; the elevation of the 
base of C = 3° 45' ; of top of do. = 
9° 25' ; of the base of D = 3° 54' ; 
of top of do. = 10° 29' 30". At B, 
the angle of position of D from C 
was = 6° 14' 30" ; and of A from C 
= 80° 51' 30", and for verification 
the elevations at B were of base of C = 3° 44', of top of 
do. = 9° 22' 15" ; of base of D = 3° 46', and of top of do. - 




116 



PLANE TRIGONOMETRY. 



[Chap. in. 



10° 7' 30". Eequirecl the heights of the trees, and the dis- 
tance between their bases. 

Ans. Height of C = 89.37 ft. ; of D = 103.37 ft. ; dis- 
tance, 1.00.7 ft. "With the angles of verification ; height of 
C = 103.29 ft.; of D = 89.36 ft. 



Fig. 77. 



Problem 13.— One side EF (Fig. 77) of a tract of land 
being inaccessible, and there beiDg no station from which 
the two ends conld be seen, I selected four 
stations, A, B, C, D ; A and D being in the 
adjoining sides, and B and C between 
them. The following measurements were 
then taken,— viz. : AB = 7.37 ch. ; BC = 
8.95 ch., and CD = 9.33 ch. ; at A, the angle 
EAB was 64° 37'; at B, ABE was 72° 43', 
and EBC 149° 32'; at C, BCF was 139° 
47', and FCD 69° 38' ; and at D, CDF was 
82° 35'. Kequired AE, EF, FD, and the 
angles AEF and EFD. 




Ans. EF = 33.50; AE = 10.38; DF = 18.77; 
AEF = 86° 39' ; EFD = 54° 29'. 



Fig 78. 



Problem 14. — Being desirous of finding the elevation and 
distance of an elevated peak C (Fig. 
78) of a mountain rising abruptly 
from the shore of a river, and not 
being able to find a level place for 
a base line, or a regular slope as- 
cending in a line from the point to 
be measured, I selected two stations, 
the one A nearly opposite the base D 
of a rock jutting into the water, and 
which was so situated that A, C, and 
D were in the same vertical plane, 
and the other station B farther up the stream, the slope 
between them being regular. I then took the following 







Sec. VI.] MISCELLANEOUS PROBLEMS. 117 

measurements, — viz. : AB, 850 yards. At A, the angle of 
position of B and C was 87° 49'; elevation of C, 35° 27'; 
depression of J), 3° 25' 45"; elevation of top of a staff at 
B of same height as the instrument, 3° 14' 30". At B, the 
angle of position of A and D was 47° 39', and of A and C, 
70° 43' 30". Depression of A, 3° 14' 30" ; of D, 4° 48' 30" ; 
elevation of C, 33° 6'. Required the horizontal distance of 
C and D from A and B, and the elevation of A, B, and C 
above the water. 

Ans. Horizontal distance of C from A, 2189.8 yds. ; from 
B, 2318.1 yds.; of D from A, 894.3 yds. ; from B, 1209.2 yds. 
Elevation of C, 1612.7 yds. ; of A, 53.6 yds. ; and of B 101.7 
yds. 



CHAPTER IV. 

CHAIN SURVEYING, 



SECTION I. 
. DEFINITIONS. 

200. Definition. Land Surveying is the art of mea- 
suring the dimensions of a tract of land, so as to furnish 
data for calculating the content and determining the area. 

201. The position of the angular points of a tract may 
be determined either by measuring the lines of the survey, 
the diagonals, offsets, &c, or by linear measures in connection 
with angular distances. These different methods of fixing 
the points give rise to different modes of surveying, — the 
first of which, as it is performed principally by the chain, 
may be called chain surveying. 

202. Advantages. As the chain, or some substitute, such 
as a tape-line or a cord, is readily procured by every one, 
surveying by this method may be performed where the 
more expensive instruments cannot readily be procured. 
To every farmer it may be important to know the content 
of a particular field, or of several fields, that he may divide 
them properly, or that he may know the value of crops 
which he is about to buy or to sell ; or for various other 
purposes that need not be mentioned. He should, there- 
fore, not be under the necessity of calling in a professional 
man to do for him what he himself, with a pair of carriage 
lines, can do, if not as well, yet fully well enough for all 
practical purposes. 

118 



Sec. II.] FIELD OPERATIONS. 119 

In order that this very simple method may be fully 
understood, we shall treat of it somewhat at length. It 
must not be inferred from this that it is recommended in 
preference to the other methods to be explained here- 
after, but only as a substitute to be used, when, from the 
circumstances of the case, these are inapplicable or incon- 
venient. 

203. Area Horizontal. It must be remembered that, 
in land surveying, it is the horizontal area that is required, 
and not the actual surface of the ground. Every measure- 
ment must, therefore, be made horizontally, as explained 
in Art. 149, et seq., and, where angles are taken, they must 
be horizontal angles. 

As the method of chaining has been fully explained in 
the articles above referred to, it will be unnecessary to 
repeat the directions here. There are, however, certain 
preliminary operations to be performed, which will form 
the subject of the next section. 



SECTION II. 

FIELD OPERATIONS. 

A.— TO RANGE OUT LINES, AND TO INTERPOLATE 

POINTS. 

204. Hanging out Lines. This requires three persons, 
each of whom should be provided with a rod some ten or 
twelve feet long, one end being pointed with iron, that it 
may be thrust in the ground. He should also have a 
plumb-line, that he may set his rod upright. The lirst, 



120 CHAIN SURVEYING. [Chap. IV. 

whom we shall call A, takes his station at the point of be- 
ginning. Looking in the direction of the line, he places B 
in the proper direction, signalling him to the right or left 
as may be required. "When the position is determined, B 
sets his rod firmly in the ground. C then goes forward, 
and looking back, by ranging with the rods of B and A, he 
puts his rod in line. A then comes forward, and, going 
ahead of C, puts himself in line, by ranging with C and B. 
They thus continue, the hindmost always coming forward, 
until the other end of the line is reached. At the point at 
which each rod was erected a stake should be driven for 
future reference; 

Lines may be prolonged in the same manner to any 
extent that may be desired. 

If the operation is carefully done, the rods being set 
plumb, the line will vary very slightly, if at all, from a 
straight line, even when extended several miles. 

205. To interpolate points in a line. The men in 

chaining should keep themselves exactly in line. This 
may readily be done by a careful follower, when the end 
of the line can be seen. If, however, one end is not visi- 
ble from the other, and from every point in the line, there 
will be nothing by which the follower can range his leader, 
unless there are staves set up for that purpose, at points 
along the line. The fixing of such points is called inter- 
polation. 

206. On level ground. If, for any purpose, such points 
were needed in a line on level open' ground, a person, 
stationing himself at one end, can signal another into the 
proper position. As many points as are wanted can thus 
be determined. 

207. Over a hill. If a hill intervenes, from the top of 
which both points may be seen, let two persons, provided 
with rods, put themselves as near in line as possible. 
Then, by alternately signalling to each other, their proper 



Sec. II.] FIELD OPERATIONS. 121 

places can be found. Thus, let XY (Fig. 79) be the Fig. 79. 
line to be interpolated. A will take his station in 
the supposed position of the line, and signal B 
until he ranges with X. B then places A in line 
with Y at C ; A again signals B to D, in line with 
X ; and so they proceed till they are both in the 
line XY. 

208. If an assistant is not at hand, or if but bi[d| 
one point can be found from which both ends of 
the line can be seen, one person can put himself 
in line by having a rule with a sight at each end ; 
wires, set upright, will do very well : lay this on 
some support, and then go to each end in turn, 
sighting to the end of the line ; he can thus deter- 
mine whether it is the proper position, and alter it until he 
finds himself rightly placed. 

209. By a Random Line. When the ends cannot be 
seen from each other, nor from any intermediate point, it is 
necessary to run a random line. This is done as directed 
in Art. 204, following a course as near that of the line to 
be interpolated as possible. 

"When the foremost person has come opposite the end of 
the line, measure the whole length, noting the distance to 
each stake, (the stakes, for convenience, being set as nearly 
as possible at equal distances ;) also measure the distance 
by which the end of the line is missed, then say : — 

As the whole distance is to the distance to any stake, so 
is the whole deviation to the correction for that stake. 
Measure the distance thus determined, in the proper di 
rection, and set the stake, or a stone, accordingly. 



122 



CHAIN SURVEYING. 




Tims, let AB (Fig. 80) be the line to be inter- 
polated. Run the random line AC, setting stakes 
at D, E, F, &c. Measure CB and the distance 
from A to D, E, F, and C. 

Suppose AC measures 27.56 chains, AD 10 
chains, AE 15 chains, AF 20 chains, and BC = 
1.57 chains. 

Then, 27.56 : 10 : : 1.57 : .57, the correction for D. 
Similarly, Ee = .85, and F/= 1.14 chains. 

Set off ~Dd, Ee, and F/, the calculated distances ; 
set stakes at d, e, and /, and range out the line 
anew. 

Instead of working out each proportion, it is 
more concise to divide the deviation by the num- 
ber of chains in the measured length : this will give the 
correction for one chain. This correction, being multi- 
plied by the distance to each stake, will give the correction 
for that stake. 

Thus, in the above example, 

— — -= .057, the correction for 1 chain. 
27.56 ' 

10 x .057 = .57, the correction for D ; 

15 x .057 = .85, the correction for E ; 

20 x .057 = 1.14, the correction for F. 

210. Across a valley. When the line runs across a 
valley, let two points A and B be determined on opposite 
sides of the valley, from which the intervening ground can 
be seen. Then let one person take his station at A, and, 
holding a plumb-line over the stake, let him sight to B : he 
can then direct his assistant into the proper position, and 
thus fix as many points as are desirable. 

Note. — These operations are all done more accurately and rapidly by means 
of the transit or theodolite. 






Sec. II.] FIELD OPERATIONS. 123 

2H. To determine the point of intersection of two visual 
lines. 

This is most readily done by three persons, two of 
whom take their stations in the lines, at some distance 
from the point of intersection, and, looking along their 
lines respectively, signal the third until he ranges in both 
lines. A stake may then be driven at the point of inter- 
section. 

This operation may readily be performed by two persons. 
First, let them run out one of the lines, and stretch a cord 
or the chain across the course of the other. One of them 
then taking his station in the second line can signal the 
other to his proper position. 

212. To run a line towards an invisible intersection. 

Through P (Fig. 81) Fig . 81 . 

run the line AC, in- 
tersecting the given 
lines in A and C. 
Then through any 
point B in*AB set out 
BD parallel to AC by c 
one of the modes to be pointed out. (See Arts. 227-229.) 
Divide BD in F, so that BF : FD : : AP : PC; that is, 

BD . AP 

make BF = — ^-- — . Then PF will be the required linei 
AC 

B.— PERPENDICULAKS. 

Problem 1. — To draw a perpendicular to a given line from a 
given point in it. 

213. (a.) When the Point is accessible. This may be 
done on the ground by the methods described in Arts. 88, 
89, and 90, using the chain for a pair of compasses to sweep 
the circles, or by the following methods : — 




D 



124 



CHAIN SURVEYING. 



[Chap. IV. 




214. First Method. Let AB Fig. 82. 

(Fig. 82) be the line and C the 
point at which the perpendicular 
is to be erected. First, lay off 
CD, 60 links; then, fixing one 
end of the chain at D, sweep an 
arc of a circle at E, using the 

whole chain (100 links) for a j— 

radius. Next, fix one end at C, 

and, with 80 links for a radius, sweep an arc cutting the 

former in E. CE will be perpendicular to AB. 

Any other distances, in the same ratio as the above, will 
answer. Thus, DC might be 30, CE 40, and DE 50. 
"With these numbers no circles need be struck. Lay off 
DC = 30 links; fix the end of the chain at D, and the end 
of the ninetieth link at C : then, taking the end of the 
fiftieth link, stretch both parts of the chain equally tight, 
and set a stake at the point of intersection. 

These numbers are very convenient when short perpen- 
diculars are required ; but when the line is run to some dis- 
tance the greater lengths are preferable. 



215. Second Method. Make AC 
(Fig. 83) a chain. With the whole 
length of the chain sweep two arcs 
cutting in D ; range out AD, making 
DE = AD : then CE will be the per- 
pendicular required. 

For, ADC being equilateral, A= 
60°,andAandACD = 120°; whence 

DCE and DEC = 60°. But DE = DC : 

therefore DCE = 30°, and ACE = 90°. 



Fig. 83. 




Sec. II.] 



FIELD OPERATIONS. 



125 



Fig. 84. 



216. (6.) When the Point is inaccessible. 

Erect a perpendicular at 
some other point D (Fig. 84) of 
the line. Through F, a point 
in this perpendicular, draw 
FH parallel to AB, (Art. 227.) 
Take FE = FD : range out EC, 
intersecting FH in G. Make 
GH equal FG: then CHI will 
be- the perpendicular required. 

FE need not be taken equal to DF. If unequal, GH will 
be determined by the proportion EF : FD : : FG : GH. 




(c.) If the line is inaccessible, trigonometrical methods 
must be employed. 

Problem 2. To let fall a perpendicular to a line from a 
point without it. 



(a.) When the point and line are both accessible. 



217. The methods in Arts. 91, 92, 
93, may be adopted in this case ; 
or in AB (Fig. 85) take any point 
D, and measure CD. Make DE = 
DC, and measure CE. 



Fig. 85. 



"*-^ C 



Then take EF = 



EC 2 



-, and F a 



• 


f 










/ 


/ 


\ \ 


/ 


/ 


'. \ 


/ 


/ 


'» \ 








/ 




\ \ 


1 


/ 


\ » 


' 


/ 


\l 



2.ED 
will be the foot of the perpendicular. 

Describe the semicircle EC A. Then, if CF is perpen- 
dicular to AB, EC is a mean proportional between AE 

EC 2 EC 2 
and EF, whence EF = — = — 



126 



CHAIN SURVEYING. 



[Chap. IV. 



Fig. 86. 




(b.) If Hie point is remote or inaccessible 

218. First Method.— In AB 
(Fig. 86) take any convenient 
» points A and D; erect the 
perpendicular FDE, making 
FD = DE; range out AE, 
and EC cutting AB in H, and 
FH intersecting AE in G: 
then GBC will be perpen- 
dicular to AB. 



For, by construction, the triangles ADE and ADF, as also FDH and EDH, are 
equal in all respects. Hence, AFG and AEC, having two angles and the included 
side of one equal to two angles and the included side of the other, are equal 
in all respects ; therefore AG = AC. Finally, ABC and ABG have two sides 
and their included angles respectively equal, whence B is a right angle. 



219. Second Method. — Select 
any two convenient stations E 
and F (Fig. 87) from which C 
may be seen, and range out FC 
and EC. To these draw the 
perpendiculars EG and FH cut- 
ting in I: then CID will be the 
perpendicular required. 




A E 



For the perpendiculars to the three sides of a triangle from the opposite 
angles intersect in the same point. 



(c.) If the line be inaccessible. 

220. From the given point 
C towards two visible points A 
and B (Fig. 88) of the given 
line range out CA and CB, 
and by one of the preceding 
methods draw the perpen- 
dicular EA and BD inter- 
secting in F : CF will be the 
perpendicular required. 



Fi<?. 88. 




221. The preceding methods will apply in all the cases 



Sec. II.] FIELD OPERATIONS. 127 

enumerated. They are, however, only to be considered as 
substitutes for the neater and more accurate methods bv 
the use of the theodolite or transit. Measurements such 
as those directed above, when they are intended to de- 
termine the direction of an important line, require to be 
made with scrupulous accuracy ; for every deviation will be 
magnified as we proceed. An error of two or three inches, 
which would be a matter of but little importance in a line 
of a chain long, would cause a deviation of from twelve to 
twenty feet if the line were prolonged to a mile. 

In the absence of a transit or theodolite, the following 
simple instruments, either of which can be constructed by 
any one having a moderate degree of facility in the use of 
tools, will enable the surveyor to lay out perpendiculars 
with readiness and considerable accuracy. 

222. The Surveyor's Cross. This consists of a block 
of wood four or five inches in diameter, with two saw-cuts 
across its centre precisely at right angles. An auger hole 
should be made at the bottom of each saw-cut, to afford a 
larger field of view. The block is fastened to the top of a 
staff' about eight or ten inches long. It should turn freely 
but firmly on the head of the staff. 

Instead of saw-cuts, four wires may be set upright at the ex- 
tremities of perpendicular diameters ; but, as these are likely 
to be deranged, the other form is better. 

223. To erect a perpendicular with the cross, set it up at 
the point at which the perpendicular is to be drawn, and 
turn it round till one of the cuts ranges with the given line ; 
then, looking through the other cut, the surveyor can direct 
his assistant to set a stake in the required perpendicular. 

If the point is out of the line, take a station as near as 
the eye can judge to the position of the foot of the per- 
pendicular, and, having set the cross so that one cut may 
range with the given line, look through the other, and see 
how far the line of sight misses the given point. Move the 
cross that distance and test it again. A few trials will de- 
termine the proper position. 



128 CHAIN SURVEYING. [Chap. IV. 

224. To verify the Accuracy of the Cross. Place it 
at a given station: range with one of the cuts to a well- 
defined object, and place a stake in the perpendicular; then 
turn the cross one-quarter round, and if the stake is in the 
perpendicular, the cross is correct, but if not, the instru- 
ment is in error by half the observed deviation. 

This will be apparent by Fig. 89. 

reference to Fig. 89. If the 
angle ACD is acute, the 
stake will be placed to the 
left of the true position, as 
at F. By turning the block 

one-fourth round, the acute a 

angle will be found at BCE, 
and the stake will be posited 
at Gf, as far to the right as it was before to the left. 

225. The Optical Square. The optical square is a much 
more convenient instrument for drawing perpendiculars 
than the cross. It consists of a circular box, having a fine 
vertical slit cut in one side, and directly opposite a circular 
or oval opening with a vertical line, such as a horsehair 
stretched across it. The box contains a piece of looking- 
glass set across it, so as to make an angle of 45° with the 
line of sight. From the upper half of this glass the sil- 
vering must be removed. Half-way between the two open- 
ings mentioned is another, to allow the rays coming from 
an object in the perpendicular to fall on the mirror and be 
reflected to the eye. 




Sec. IL] 



FIELD OPERATIONS. 



129 



Fig. 90 represents a Fig. 90. 

plan of this instru- 
ment. ABC is a sec- 
tion of the box, A the 
slit at which the eye is 
placed, B the opening 
in the line of sight, 
C the opening for the 
perpendicular, and DE 
the looking-glass. 

The surveyor holds 
the box in his hand, 
and, looking at the other end of the line, through the open- 
ings A and B, directs his assistant, who is seen by reflec- 
tion through C, to place his rod in such a position that its 
image shall coincide with the hair across the opening B. 
HG- is then perpendicular to AF. 

To find the point in which the perpendicular from a dis- 
tant point will intersect AF, walk along the line, keeping 
the line of sight AB directed to the end of the line. When 
the image of a pole standing at the point from which the 
perpendicular is to be drawn appears at H, the proper posi- 
tion has been attained. 




226. To test the Accuracy of the Square. Erect a 
perpendicular with it, as above directed. Then sight along 
the perpendicular, and if the original line appears perpen- 
dicular, the instrument is correct ; if it does not, the devia- 
tion will equal twice the error of the instrument. Set a 
pole in the true perpendicular, which will be found as in 
Art. 224, and alter the position of the glass until the re- 
flected image appears in the proper position. One end of 
the glass should be movable by screws or by little wedges, 
so as to allow of its position being rectified. 



130 



CHAIN SURVEYING. 



[Chap. IV. 



C— PARALLELS. 

Problem 1. — Through a given point to run a parallel to a 

given accessible line. 



227. This may be done by Arts. Fig. 91. 
97, 98, or 99, or thtfs :— * »- 7 F 2 

Let AB (Fig. 91) be the line, and \ ; {' 

C the point. From C to any point / \ 

D in AB, run out the line CD. -c- _i> 

From E, any point in CD, run a 

line cutting AB in F. Then make EG a fourth proportional 

EF.EC 



toDE,EF,andEC,orEG 
lei to AB. 



ED 



-, and GC will be paral- 




Froblem 2. — To draw a parallel to an inaccessible line, two 
points of which are visible. 



228. Let AB (Fig. 92) be the 
straight line, and C the given 
point. Run the line CD per- 
pendicular to AB, by Art. 220 ; 
and from C set out CE perpen- 
dicular to CD. It will be the 
parallel required. 



Problem 3. — To draw a parallel to a given line through an 
inaccessible point. 

229. Let AB (Fig. 
93) be the given line, 
and C the given point. 
From A, towards C, 
run AC; and in CA, 
or CA produced, take 
any point D. Run DE 
parallel to AB. Set 
off BC towards C, in- 




Sec. III.] OBSTACLES IN RUNNING AND MEASURING LINES. 131 

tersecting DE in E. Measure AB and DE. Run through 

any point in AB the line BFG, intersecting DE in F. 

DE.BF 
Make FG = — - — =—3 and CG will be parallel to AB. 
AB — DE 

DE . BF 

For, since FG = — - — — - >wehaveAB-DE:DE::BF:FG. 
AB — DE 

Whence AB : DE : : BG : FG; 

but AB : DE : : BC : EC ; 

BG : FG : : BC : EC, and CG is parallel to EF, or 
to AB. 



SECTION III. 

OBSTACLES IN RUNNING AND MEASURING LINES.* 

A.— OBSTACLES IX RUNNING LINES. 

230. In ranging out lines by the method described in 
Art. 204, obstacles are frequently met with which prevent 
the operation being directly carried on. In such cases 
some contrivance is necessary in order that the line may be 
prolonged beyond such obstacle. Various methods have 
been devised for this purpose. The 'following are among 
the most simple : — 

231. First Method. — By per- Fig. 94. 
pendiculars. Let AB (Fig. 94) 
be the line, and M the obsta- — ■ 
cle. At two points C and B 
in AB, set off two equal per- 
pendiculars CD and BE long enough to pass the obstacle. 
Through D and E run the line DG ; and at two points F 
and G beyond the obstacle, set off perpendiculars FH 

* In Gillespie's "Land Surveying" may be found a still greater variety of 
methods for these objects. 




K 



182 



CHAIN SURVEYING. 



[Chap. IV. 



and GI equal to CD. Then HIK will be the prolongation 
of AB. 



A B 




232. Second Method.— By 
equilateral triangles. Let AB 
(Fig. 95) be the line, the 
obstacle being at 0. By 
sweeping with the chain, 
describe the equilateral tri- 
angle BCD. Prolong BD 
to E sufficiently far to pass 
the obstacle. Describe the 
equilateral triangle FEG, and prolong EG till EH = EB. 
Describe the equilateral triangle HKI, and KH will be the 
prolongation of AB." 

233. Instead of making BEH an equilateral triangle, 
which would sometimes require the point E to be incon- 
veniently remote, run BE (Fig. 
96) as before. Set out the per- 
pendicular EG = 1.T32 x BE. 
Describe the equilateral triangle 
GFL Bisect FI in H. Then 
HG will be the prolongation 
of BC. 



Fig. 96-j. 




B .— OBSTACLES IN MEASURING^ LINES. 

234. When, owing to any obstructions, the distance of a 
line cannot be directly measured, resort should be had to 
trigonometrical methods. In the absence, however, of the 
proper instruments, it may be necessary to determine such 
distances. The following are a few of the many methods 
that may be employed in such cases : — 

1. To measure a line when both ends are accessible. 

235. Arts. 231, 232, 233, furnish means of determining 
the distance in this case. By the method Art. 231, BH = 






Sec. Ill,] OBSTACLES IN RUNNING AND MEASURING LINES. 133 

EF ; and in that of 232, BH = BE. If the method Art. 233 
is employed, BG = 2 BE. 

2. When one end is inaccessible. 



236. First Method.— Rim BE (Fig. 97) 

in any direction, and AD parallel to it. 

Through any point D in AD, run DE 

towards C. Measure AD, AB, and BE : 

AB.BE 

then BC = aU^be 



Fig. 97. 




237. Second Method.— Set off AC (Fig. 
98) in any direction, and CD parallel to 
AB. Eun DE towards B. Measure AE, 

AE.CD 



EC, and CD : then AB = 



CE 



Fig. 98. B 




238. Third Method.— Set off AD (Fig. 
99) perpendicular to AB, and of any dis- 
tance. Bun DC perpendicular to DB. 

CD 2 
Measure DC and CA : then CB = 



Fig. 99 



CA' 



or AB = 



AD 2 
ACT 




3. When the point is the intersection of the line with another, 
and is inaccessible. 



134 



CHAIN SURVEYING. 



[Chap. IV. 



239. First Method.— Let 

AB and CD (Fig. 100) be 

the lines, the distances of 

which to their intersection 

are required. Set off DF 

parallel to BA, and run 

CFA. Measure CD, CF, 

CA,andFD. Then BE = 

BD.DF lT _ BD.DC 
-jandDE = 



Fig. 100. 





FC 



CF 



240. Second Method.- 
in CD, run two lines 
AF and BG. Make 
FH in any ratio to HA, 
and GH in the same 
ratio to HB. Draw 
FGC, cutting CD in 
C. Measure FC and 
HC. Then AE = 

AH.FC 

and HE = 



-Through H, (Fig. 101,) any point 

Fig. 101. 



FH 
AH.HC 




FH 
4. When both ends are inaccessible. 

241. Let AB (Fig. 102) be the in- 
accessible line. From any con- 
venient point C, run the lines CA 
and CB towards A and B, and, by 
one of the preceding methods, find 
CA and CB. In CA and CB, or 
CA and CB produced, take E and D 

so that CE : CA : : CD : CB. 

Measure DE. 



Fig. 102. 




Then 



CE : CA : : ED : AB. 



D- 



Sec. IV.] KEEPING FIELD-NOTES. 135 



SECTION IV. 

KEEPING FIELD-NOTES. 

242. The operation next in importance to that of per- 
forming the measurements accurately is that of recording 
them neatly, concisely, and luminously. The first is a 
requisite that cannot be too much insisted on, not only 
in the first notes, but in all the calculations and records 
connected with surveying. A rough, careless mode of re- 
cording observations of any kind generally indicates an 
equal carelessness in making them. Carelessness in a sur- 
veyor, on whose accuracy so much depends, is intolerable. 
Conciseness is also necessary, but it should never be al- 
lowed to detract from the luminousness of the notes. By 
this last quality is meant the recording of all the observa- 
tions in such a mode as to indicate, in the most clear man- 
ner, the whole configuration of the plat surveyed, and all 
the circumstances connected with it which it is intended to 
preserve. The notes should be, in fact, a full record of all 
the work, so as to indicate fully not only what was done, 
but what was left undone. 

243. First Method. — By a sketch. The simplest mode of 
recording the notes is to draw a sketch of the tract to be 
surveyed, on which other lines can be inserted as they are 
measured. On this sketch may be set down the distances 
to the various points determined. 

When the tract is large, however, or contains many base- 
lines, this sketch becomes so complicated as scarcely to be 
capable of being deciphered after the mind has been with- 
drawn from that particular work and the configuration of 
the plat has been in some measure forgotten. 

244. Field-Book. Perhaps the best kind of a field- 
book is one that is long and comparatively narrow, faint- 
lined at moderate distances. The right-hand page should 



136 CHAIN SURVEYING. [Chap. IV. 

be ruled from top to bottom with two lines, about an 
inch, apart, near the middle of the page. The left-hand 
page may be ruled in the same manner ; but it is better 
left for remarks, sketches, and subsidiary calculations. 

In the space between the vertical lines all the distances 
are to be inserted: offsets, and other measurements con- 
nected with the main line, may be recorded in the spaces on 
each side of the column. 

In recording the measurements the book should be held 
in the direction in which the work is proceeding. The 
right-hand side of the column will then coincide with the 
right-hand side of the line, and vice versa. The notes 
should commence at the bottom, and all offsets and other 
lateral distances must be recorded on the side of the 
columns corresponding to the side of the line to which 
they belong. 

When marks are left for starting points for other mea- 
surements, the distance to them should be recorded in the 
column, and some sign should be made to indicate the 
purpose for which such distance was recorded. Stations 
of this kind are called False Stations, and may be desig- 
nated by the letters F. S. ; by a triangle, a ; or circle, o ; 
or by surrounding the number by a circle, thus, f 567. ) 
Whatever plan is adopted should be scrupulously adhered 
to, — changes in the notation beiug always liable to lead to 
confusion. 

A regular station may be designated either by letters, A, 
B, or by numbers, 1, 2, 3, prefixed by the letter S or by Sta. 
In the field-notes in the following pages examples of most 
of these methods will be found. 

Lines are referred to, either by having them numbered 
on the notes as Line 1, Line 2, or by the letters or figures 
which designate the stations at their ends. Thus, a line 
from Sta. 1 to Sta. 3 would be referred to as the line 1, 3 ; 
one from Sta. B to Sta. D, as the line BD. This is perhaps 
the best mode. Some surveyors, however, refer to them by 
their lengths. Thus, a line 563 links long would be called 
the line 563. 

False stations on a line are named by the line and distance. 



Sec. IV.] 



KEEPING FIELD-XOTES. 



137 



Thus, a station on a line AB at 597 links would be called 



F. S. 597 AB, or (597 ) AB, or A, or O 597 AB. It hardly 
needs remark, yet it is of importance, that unity of system 
should be adopted. Whatever method of designating a 
line or station has been employed in recording it, should be 
used in referring to it. 

The spaces on the right and left of the column will serve, 
in addition to the purposes already mentioned, to contain 
sketches of adjoining lines and short remarks to elucidate 
the work. 

A fence, road, brook, &c. 
crossing the line measured, 
should not be sketched as 
crossing it in a continuous 
line, as at 365, marginal 
plan, but should consist of 
two lines starting at opposite points, as at 742, so that if we 
were to suppose the lines forming the vertical column to 
collapse, those representing the fence would be continuous. 

When the chainmen, after closing the work on one line, 
begin the next at the closing station, a single horizontal 
line should be drawn; but if they pass to some other part 
of the tract, two lines should indicate the end of the line. 

To indicate the direction in which a line turns, the marks 
1 or T raay be used, the former indicating that the new 
line bears to the left, and the latter to the right. Instead 
of these, the words right and left may be used, or the simple 
initials E. and L. Whichever of the means is used, the 
sign should be on the left hand of the column if the turn is 
to the left, and vice versa. 



Sta. B 




947 




742 




_3££- — ' 




ri27 


F. S. 


Sta. A 


K 15° E. 



The following notes will illustrate all these directions: 
They belong to the tract Fig. 103. 



1.38 



CHAIN SURVEYING. 



[Chap. IV. 



1 


Sta. D 
2440 
2020 

(1395) 
Sta. A 


"^ 


1 


Sta. A 

1135 

Sta. C 




^~^T 


Sta. C 

1760 

( 950) 

Sta.B 




^'- 


Sta. B 
2492 
1445 
1170 

Sta. A 


N.45°E. 




^ Brook. 











(1395) 


in AD 


"*--^^ 


1440 






770 


-"" 




425 


^•"**"Ci 


^■^ 


( 950) 


In BC sout 




Sta.B 






1760 


,Bi 


s^~ 


515 




^ -\ 


Sta.D 





Brook. 




Beginning at A, the first line measured is the diagonal 
AB ; the course N. 45° E. is set down at the right. The 
first point requiring notice is the intersection of the dia- 
gonals at 1170 links from A. The diagonal is represented 
by the dotted line crossing the columns, a continuous line 
being employed to designate a fence or side, and a dotted 
line a sight-line. At 1445 the fence EF is crossed. The 
whole length to B is 2492 links. 



Sec. IV.] KEEPING FIELD-NOTES. 139 

Turning to the left along BC ; at 950 we come to the fence 
bearing to the left: 950 is surrounded by a line, thus, f 95(A 
because it is to be used as a starting-point for another mea- 
surement. Having arrived at C, 1760 links from B, again 
turn to the left towards A: the distance CA is 1135 links. 
AD is next measured. At 1395 the fence EF is found : the 



point is marked (1395 j : at 2020 the brook is crossed, and 
at 2440 links we find the corner D. Turning to the left 
along DB, at 515 the brook is again crossed. This line is 
1760 links long. 

Passing now to E, ( 950 J in BC, along the cross fence, 
the diagonal AB is passed at 425; at 770 CD is passed; 
1440 links brings us to 1395 in AD. Passing to-D: along 
DC, at 395 the brook is crossed ; at 1390 the fence is found ; 
at 1550 we cross the diagonal AB: 2425 brings us to C, 
which finishes the work. 

245. Test-lines. In the above survey more lines have 
been measured than are absolutely necessary. It is always 
better to measure too many than too few. If the redundant 
lines are not needed in the calculation, they serve as tests by 
which to prove the work. For the mere purpose of calcula- 
tion, one of the diagonals and the line EF might have been 
omitted : the other lines afford sufficient data for making a 
plat and calculating the area. An error in one of the others 
will not prevent the notes from being platted, and hence 
they do not in any way afford a criterion by which we can 
judge of the accuracy of the measurements; but when to 
these are added the length of the other diagonal we have a 
series of values, all of which must be correct or the map 
cannot be made. 

246. General Directions. When about to survey a 
tract by this method, the surveyor should first examine the 
tract carefully and erect poles at the prominent points, 
corners, and false stations, along the boundary lines. He 
should stake out all diagonals and subsidiary lines which 
he may wish to measure, setting a stake at the points in 



140 CHAIN SURVEYING. [Chap. IV. 

which such lines intersect each other or cross the former 
lines, — in fact, at every point the position of which it may 
be desirable to fix on the plat. 

Having made these preparations, he may, if the tract is 
at all complicated, make an eye-sketch. This will serve to 
guide him in regard to the best course to take in his 
measurements. 

Commencing then at some convenient point of the tract, 
he should measure carefully the diagonals and sides in suc- 
cession, passing from one line to such other as will make 
the least unnecessary walking, and setting down in his note- 
book the distance to every stake, fence, brook, or other im- 
portant object met with. 

When the tract is large, the work may last through 
several days. In such cases, each day's work should, if 
possible, be made complete in itself, — that it may be platted 
in the evening. This will prevent the accumulation of 
errors which might occur from a mismeasurement of one 
of the earlier lines. 

247. Platting the Survey. To plat a survey from the 
notes, select three sides of a triangle and construct it. 
Then, on the sides of this construct other triangles, until 
the whole of the lines are laid down. Measure test-lines to 
see whether the work is correct. 

In all cases commence with large triangles, and fill up 
the details as the work proceeds. 



Sec. Y.] SURVEYING FIELDS OF PARTICULAR FORMS. 141 



SECTION V. 

ON THE METHOD OF SURVEYING FIELDS OF PAR- 
TICULAR FORMS. 

248. Rectangles. Measure two adjacent sides: their 
product will give the area. 

Examples. 

Ex. 1. Let the adjacent sides of a rectangular field be 
756 and 1082 links respectively, to plat the field and calcu- 
late the content. 

Calculation. 

Content = 1082 x 756 = 817992 square links = 8 A., OR., 
28.7 P. 

Ex. 2. The adjacent sides of a rectangular tract are 578 
and 924 links : required the area. 

Ans. 5 A., IE., 14.51 P. 

Ex. 3. Required the area of a tract the sides of which 
are 9.75 and 11.47 chains respectively. 

Ans. 11 A., R., 29 P. 

249. Parallelograms. Measure one side and the per- 
pendicular distance to the opposite side. Their product 
will be the area. 

If a plat is required, a diagonal or the distance from one 
angle to the foot of the perpendicular let fall from the adja- 
cent angle may be measured. 

Examples. 

Ex. 1. Given one side of a parallelogram 10.37 chains, 
and the perpendicular distance from the opposite side 7.63 
chains, the distance from one end of the first side to the 
perpendicular thereon from the adjacent angle being 2.75 
chains. Required the area and plat. 

Ans. 7 A., 3 R., 25.97 P. 



x 42 CHAIN SURVEYING. [Chap. IV. 

Ex. 2. Desiring to find the area of a field in the form of 
a parallelogram, I measured one side 763 links, and the 
perpendicular from the other end of the adjacent side 647 
links, said perpendicular intersecting the first side 137 links 
from the beginning. Required the content and plat. 

Ans. 4 A., 3 R., 29.86 P. 

250. Triangles. First Method. — Measure one side, and 
the perpendicular thereon from the opposite angle ; noting, 
if the plat is required, the distance of the foot of the per- 
pendicular from one end of the base. 

Multiply the base by the perpendicular, and half the pro- 
duct will be the area. 

* 

Examples. 

Ex. 1. Required the area and plat of a triangular tract, 
the base being 7.85 chains and the perpendicular 5.47 chains, 
the foot of the perpendicular being 3.25 chains from one 
end of the base. 

Calculation. 

7.85x5.47 42.9395 Mi MM . . 
Area = = = 21.46975 chains = 2 A., 

2 2 ' 

R., 23.5 P. 

Ex. 2. Required the area and plat of a triangle, the base 
being 10.47 chains, and the perpendicular to a point 4.57 
chains from the end, being 7.93 chains. 

Ex. 3. Required the area of a triangle, the base being 
1575 links, and the perpendicular 894 links. 

251. Second Method. — Measure the three sides, and calcu- 
late by the following rule: — 

From half the sum of the sides take each side severally ; mul- 
tiply the half-sum and the three remainders continually together, 
and the square root of the product will be the area. 



Sec. V.] SURVEYING FIELDS OF PARTICULAR, FORMS. 



143 



Demonstration. — Let ABC (Fig. 104) be Fig. 104. 

a triangle. Bisect the angles C and A by 
the lines CDH and AD, cutting each other 
in D. Then D is the centre of the inscribed 
circle. Join DB, and draw DE, DF, and 
DG perpendicular to the three sides. Then 
will DE = DF = DG, and (47.1) FB = BG, 
CE = CF, and AE = AG. 

Bisect the exterior angle KAB by the 
line AH, cutting CDH in H. Draw HK, 
HL, and HM perpendicular to CA, AB, 
and CB. Join HB. Then (26.1) KH = 
HM, CK = CM, HL = HK, and AL == AK ; 
also (47.1) BL = BM. Because AK = AL 

and BM = BL, CK -|- CM will be equal to the sum of the sides AB, AC, and 
BC ; therefore CK or CM = £ (AB -f AC + BC) = J S, if S stand for the 
sum of the three sides. But CE -f AE -f- BG = \ S ; therefore CK = CM = 
CA + BG, and AK = AL = BG; whence AG = AE = BL = BM, and EK == 
AB. Now, since CK = CM = \ S, we have AK = \ S — AC, EC = \ S — AB, 
and AE = BM = \ S — BC. 

Because the triangles CDE and CKH, as also ADE and HKA, are similar, 




we have (4.6) 
and 

(23.6) 

Whence, 

and 



CE : ED 
AE : ED 
AE . EC : ED 2 



v/AE . EC : ED : 



CK : KH, 
HK : KA, 
CK : KA : : CK a : CK . KA. 



CK : ^/CK . KA, 



CK . ED = v'CK . KA . AE . EC. 



Now, ABC = ACD+ BCD + ABD == \ AC . ED+ \ BC . ED + \ AB . ED 
= JS . ED = CK. ED. 



Wherefore, ABC = y/CK . KA. AE . EC. 

Cor. — From the above demonstration, it is apparent that the area of a tri- 
angle is equal to the rectangle of the half-sum of the sides and the radius of 
the inscribed circle. 

For another demonstration of this rule, see Appendix. 

Examples. 

Ex. 1. Required the area of a triangle, the three sides 
being 672, 875, and 763 links respectively. 



Note. — In cases of this kind the operation will be much facilitated by using 
logarithms. 



144 CHAIN SURVEYING. 

672 + 875 + 763 2310 



[Chap. IV. 

= 1155 = half-sum of sides. 



J sum = 1155 

| sum - 672 = 483 
J sum - 875 = 280 
J sum - 763 = 392 



Area, 247449 square links, 
= 2 A., 1 E., 35.9 P. 



log. 3.062582 
log. 2.683947 
log. 2.447158 
log. 2.593286 
2) 10.786973 
5.393486 



Ex. 2. Required the area of a triangular tract, the sides 
of which are 17.25 chains, 16.43 chains, and 14.65 chains 
respectively. Ans. 11 A., R., 14.4 P. 

Ex. 3. Given the three sides, 19.58 chains, 16.92 chains, 
and 12.76 chains, of a triangular field: required the area. 

Ans. 10 A., 2 R., 27 P. 

252. Trapezoids. Measure the parallel sides and the per- 
pendicular distance between them. 

If a plat is desired, a diagonal, or the rig.105. 

distance AE, (Fig. 105,) may be mea- 
sured. 

Multiply the sum of the parallel sides by 
half the perpendicular : the product is the area. 

Demonstration. — ABCD = ABD + BCD = \ AB . DE + \ DC . DE = 
(AB + DC) . \ DE. 

Examples. 

Ex. 1. Given AB = 7.75 chains, DC = 5.47 chains, and 
DE = 4.43 chains, to calculate the content and plat the 
map, AC being 7.00 chains. 

. Ans. Area, 2 A., 3 R., 28.5 P. 

Ex. 2. Given the parallel sides of a trapezoid, 16.25 chains 
and 14.23 chains, respectively : the perpendicular from the 
end of the shorter side beinsr 12.76 chains, and the distance 




Sec. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 145 

from the foot of said perpendicular to the adjacent end of 
the longer side 1.37 chains. Required the area and plat. 

Ans. 19 A., 1 E., 31.4 P. 

253. Trapeziums. First Method.' — Measure a diagonal, 
and the perpendiculars thereon, from the opposite angle. 

The area of a trapezium is equal to the rectangle of the 
diagonal and half the sum of the perpendiculars from the 
opposite angles. 

This is evident from the triangles of which the trapezium 
is composed. 

Examples. 

Ex. 1. To plat and calculate the area of a trapezium, the 
diagonal being 15.63 chains, and the perpendiculars thereto 
from the opposite angles being 8.97 and 6.43 chains, and 
meeting the diagonal at the distances of 4.65 and 13.23 
chains. Ans. Area, 12 A., R., 5.6 P. 

Ex. 2. Given (Fig. 106) AC = 19.68 

chains, AE = 7.84 chains, AF = 16.23 

chains, ED = 10.42 chains, and FB = 

8.73 chains, to plat the figure and find 

the area. 

Ans. 18 A., 3 R., 14.98 P. 

Ex. 3. Required the area of a trape- 
zium, the diagonal being 17.63 chains, and the perpen- 
diculars 6.47 and 12.51 chains respectively. 

Ans. 16 A., 2 R., 36.94 P. 

254. Second Method. — Measure one side, and the perpen 
diculars thereon from the extremities of the opposite side, 
with the distances of the feet of these perpendiculars from 
one end of the base. 



10 




146 



CHAIN SURVEYING. 



[Chap. IV. 



Examples. 



Fig, 107. 
C 




Ex. 1. Let ABCD (Fig. 107) 
be a trapezium, of which the fol- 
lowing dimensions are given, — 
viz. : AE = 3.27 chains, AF = 
10.17 chains, AB = 17.62 chains, 
ED = 7.29 chains, and EC = 
13.19 chains. Required to plat 
it, and calculate the area. 

Lay off the distances AE, AF, and AB ; then erect the 
perpendiculars ED and FC, and draw AD, DC, and CB. 

The trapezium is divided into two triangles and the 
trapezoid, the areas of which, may be found by the pre- 
ceding rules. 

Thus, 2AED=* AE.ED = 23.8383 

2 EFCD = EF . (ED + FC) = 141.3120 

2 CFB = CF. FB = 98.2655 

whence ABCD = J of 263.4158 = 131.7079 

chains = 13 A., OK., 27.3 P. 

If either of the angles A or B were obtuse, the perpen- 
dicular would fall outside the base, and the area of the 
corresponding triangle should be subtracted. 

Ex. 2. Plat and calculate the area of a trapezium from 
the following field-notes : — 



perp. 936 
perp. 825 




Ans. 7 A., R., 30.3 P. 

Ex. 3. Calculate the area from the following field- 
notes : — 



perp. 892 
perp. 568 

Ans. 6 A., 2 R., 2 P. ~~ 




Stat. B. 



Sec. V.] SURVEYING FIELDS OF PARTICULAR FORMS, 



147 



Fields of more than four sides, bounded by 
straight lines. 

255. First Method. — Divide the tract into triangles and 
trapeziums, and calculate the areas by some of the pre- 
ceding rules. In applying this method, as many of the 
measurements as practicable should be made on the 
ground ; the field then being platted with care, the other 
distances may be measured on the map. "When it is 
intended to depend on the map for the distances, every 
part of the plat should be laid down with scrupulous ac- 
curacy, on a scale of not less than three chains to the 
inch. 

Ex. 1. To draw the map and calculate from the follow- 
ing field-notes the area of the pentagonal field ABODE : — 





OD 






C 




© c 




"3 


690 




*<§ 


770 


<6 


915 




a 
o 
fcu 


570 


510 C 


a 

o 
fcfi 


510 


250 B | 


585 


Brook. 


a E. 350 


280 




e3 

s 


360 


Brook. ^ 


365 


AD 




©A 


N.15°E. 




©A 


E. of AD 


© E 





The construction is plain. 

Calculation. 
Twice trapezium ACDE = AD 
x (Ea + bC) = 6.90 x 8.60 = 
59.34; twice triangle ABC = 
AC x Be = 7.70 x 2.50 = 19.25; 
59.34 + 19.25 



Fig. 108. 



whence ABCDE = 



2 




= 39.295 ch. = 3 A., 3 R., 28.72 P. 



Ex. 2. Map the plat, and calculate 
the content from the following field- 
notes : — 



Fig. 109. 




148 



CHAIN SURVEYING. 



[Chap. IV. 





0D 






520 






288 


80 E 


G120 


206 
o F 






©g 






440 




D230 


150 






©C 


Lof CA 




©c 






550 




B180 


410 






135 


130 G 




©A 


East. 



Construction. 

Commencing at A, (Fig. 109,) draw the line AC east 
5.50 chains, marking the points a and b at 1.35 and 4.10 
chains respectively : at a and b erect the perpendiculars aG- 
1.30 and bB 1.80 chains. From C to G draw CG, which 
should be 4.40 chains long. At c, 1.50 chains from C, 
draw cD perpendicular to CG and equal to 2.30 chains. 
With the centre G and radius 1.20 chains, describe a circle, 
and from D draw the line DF 5.20 chains long, touching 
the circle at e, which should be 2.06 chains from F. At d, 
2.88 chains from F, draw the perpendicular dE = .80 chains: 
then will ABCDEFGbe the corners of the tract. 



Calculation. 

2 ABCG = AC (Ga + Bb) = 5.50 x 3.10 = 17.05; 
2 GCD = GC . cD = 4.40 x 2.30 = 10.12; 

2 GDEF = FD (Ge + dE) = 5.20 x 2.00 = 10.40. 



Therefore area = 
3 K., 20.56 P. 



37.57 



chains = 18.785 chains = 1 A., 



Ex. 3. Required the plans and areas of the adjoining 
fields, of which the following are the field-notes, the two 
fields to be platted on one map. 



Sec. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 



149 



(3) 772 




284 (5) 

KE. 



(2) 395 




715 (6) 
K 10° E. 



Area 10 A., 2 E., 18.576 P. 





© 7 






1150 






675 

0(8) 


432 (11) 




0(9) 

1285 




(8) 565 


1000 






960 
0(7) 


155 (10) 
L. of (7,5) 




0(7) 
1315 




(4) 562 


390 

282 


313 (10) 




0(5). 


E. of (4) 



Area 12 A., 3 E., 18.1 P. 

Ex. 4. Eequired the plan and areas of the adjoining fields 
from the following field-notes, tracing thereon the course 
of the brooks. 





0(7) 






1051 




Brook + (6.7)— 


680 




-N^^ 


648 


540 (1) 


V ^^ 


510 


->« v _^ Brook. 




365 


—Brook + (1.5) 


(6) 380 


130 






0(5) 


r 




0(5) 






1255 






853 


765 (1) 


(4) 500 


440 






0(3) 


r 




0(3) 






1150 




Brook + (2.3)— 


850 






490 


^u Brook. 


(2) 482 


200 

0(1) 






Area 14 A., 3E., 28.24 P. 



Area 15 A., 2 E., 7 P. 



Note. — In the above field-notes the marginal references, such as "Brook 4 
6.7," means to the point in -which the brook crosses the line (6.7.) 



150 



CHAIN SURVEYING. 



[Chap. IV. 



Fig. 110. 



256. Second Method. — Instead of running diagonals, it may 
sometimes be more convenient to run one or more lines 
through the tract and take the perpendiculars to the several 
angles, as in the following example. 

Let the field be of the form 
ABCDEF, (Fig. 110.) Run the line 
AC, and take the perpendiculars /F, 
eE, 6B, anddD. The field will thus 
be divided into triangles and trape- 
zoids, the area of which may be 
calculated by the preceding rules. 

Thus, let the field-notes of the preceding tract be as 
follows : — 






C 






1185 




D420 


840 






760 


200 B 


E280 


590 




F220 


250 






©A 


East. 



Dist. 





250 

590 

840 

1185 



1185 x 200 





Int. 


Sum of 


Double 


Perp. 


Dist. 


Perp. 


Areas. 




220 


250 


220 


55000 


280 


340 


500 


170000 


420 


250 


700 


175000 





345 


420 


144900 



2 AF/ 
2/FE* 
2eEDd 
2BdC 



544900 
237000 



Left-hand areas. 
Eight " " 

2) 781900 
39.0950 ch. = 3 A., 3 E., 25.5 P. 



The calculation being performed thus: — In the first 
column are placed the distances to the feet of the left-hand 
perpendiculars. In the second the perpendiculars them- 
selves. The numbers in the third column are found by 
subtracting each number in column 1 from the succeeding 
number in the same column. The numbers in column 3 



Sec. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 



151 



therefore represent the distances A/, fe, ed, and dC. The 
numbers in the fourth column are found by adding each 
number in column 2 to the succeeding number in the 
same column; they therefore are the sums of the adjacent 
perpendiculars. Those in the fifth column are found by 
multiplying the corresponding numbers in columns 3 and 
4. They therefore are the double areas of the several 
trapezoids and triangles. 

Ex. 2. Required to calculate the content and make plats 
from the following field-notes : — 





0G 






312T 






2590 


476 F 


H375 


2145 






2070 


642 E 


1400 


1920 






1485 


523 D 




840 


516 C 


K600 


790 






200 


465 B 




©A 


E. 





©F 






4025 






3617 


792 G- 




3254 


826 H 


E594 


2846 




D435 


2137 






1548 


319 1 


C729 


1026 






429 


623 K 


B237 


175 






©A 


K 15° E. 



Area 25 A., 1 



E.,5P. 



Area 38 A., 3R, 17.5 P. 



257. Offsets. In what precedes, the sides have been sup- 
posed to be right lines. This is ordinarily the case except 
when the tract bounds on a stream. It then, if the stream 
is not navigable, generally takes in half the bed. Lands 
bounding on tide-water go to low-water mark. In all such 
cases the area and configuration of the boundary are most 
readily determined by offsets. 

A base is run near the crooked boundary, and perpen- 
dicular offsets are taken to the line, whether it be the middle 
of the stream or low-water mark. If the positions of these 
offsets are judiciously chosen, so that the part of the boun- 
dary intercepted between any two consecutive ones is nearly 
straight, the correct area may be calculated precisely as in 
last article. In the field-notes the distances are written in 
the column and the offsets on the right or left hand, accord- 
ing as they are to the right or left of the line run. 






152 



CHAIN SURVEYING. 



[Chap. IV. 



Thus, supposing it were 
required to find the area 
contained between the line 
AB and the stream, (Fig. 
Ill,) the following being 
the field-notes. 



Fig. 111. 






©B 




25 


865 




70 


725 




165 


580 




165 


475 




100 


355 




115 


195 




90 


75 




40 









©A 


K10°E. 



The calculation would be as below, the same formula 
being used as in last article. 



Dist. 


Offs. 


Int. 
Dist. 


Sum of 
Offs. 


Double 
Areas. 





40 








75 


90 


75 


130 


9750 


195 


115 


120 


205 


24600 


355 


100 


160 


215 


34400 


475 


165 


120 


265 


31800 


580 


165 


105 


330 


34650 


725 


70 


145 


235 


34075 


865 


25 


140 


95 


13300 


2) 182575 


Area = 3 R, 26 P. 9.12875 ch. 



Sec. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 



153 



Ex. 1. Kequired the area and plan from the following 
notes : — 



*N 


A 






4830 




\ 


2040 


***»«»^ 




F 


r "^ 




F 






21T5 




E355 


1428 






D 


r 




D 


on creek-bank 




41T5 




C665 


3335 




55 


(2160) 


B 


270 


1929 




396 


1408 




310 


1015 




340 


610 




50 









A 


K56|°E. 





E 


60 


14T1 


95 


930 


140 


485 


60 







D 





D 


60 


1072 


130 


750 


85 


390 


55 







C 



55 
55 

1 


C 

1350 



(2160) 





D 






5000 


y' 


y 


3585 


y 


s 


G 





B on A.D 





A 






3000 






G 


r 




G 






4241 






F 


r 




F 




75 


826 




100 


420 




60 







). 


E 


r 



Fig. 112 is a plat of this tract. 




154 





CHAIN SURVEYING. 


[Chap. IV. || 




Calculation. 


1 




To find AGF, Art. 251. 


1 


AG 


3000 


1 


FG 


4241 


1 


FA 


4830 

2)12071 




| sum 


6035.5 


3.780713 I 


J s - AG 


3035.5 


3.482230 J 


J s - FG 


1794.5 


3.253943 1 


J5 -AF 


1205.5 


3.081167 1 




2)13.598053- 


AGF 


= 6295435 

To find AFD. 


6.799026 


AF 


4830 




AD 


4175 




FD 


2175 






2)11180 


J sum 


5590 


3.747412 


Js-AF 


760 


2.880814 


Js -AD 


1415 


3.150756 


Js-FD 


3415 


3.533391 




2)13.312373 


AFD 


= 4530917 


6.656186 



f.] SURVEYING FIELDS OF PARTICULAR FORMS. 




To find BCD. 




BC 


1350 




BD 


•2015 




CD 


10T2 
2)4437 




J sum 


2218.5 


3.346059 


Js-BC 


868.5 


2.9387T0 


|s-BD 


203.5 


2.308564 


Js -CD 


1146.5 


3.059374 




2)11.652767 



155 



BCD = 670475 5.826383 





To find DEF. 




DE 


1471 




EF 


826 




DF 


2175 

2)4472 




J sum 


2236 


3.349472 


|s -DE 


765 


2.883661 


J 5 -EF 


1410 


3.149219 


is-DF 


61 


1.785330 




2)11.167682 



DEF = 383567 5.583841 



156 



CHAIN SURVEYING. 



[Chap. IV. 



Base. 


Dist. 


Offsets. 


Inter. 
Dist. 


Sum of 
Offsets. 


Double 
Areas. 


AB 
BC 



610 
1015 

1408 
1929 
2160 


50 
340 
310 
396 
270 

55 


610 
405 
393 
521 
231 


390 
650 
706 
666 
325 


237900 
263250 

277458 

346986 

75075 






1350 


110 


148500 


CD 




390 
750 

1072 


55 

85 

130 

60 


390 
360 
322 


140 
215 

190 


54600 
77400 
61180 


DE 




485 

930 

1471 


60 

140 

95 

60 


485 
445 
541 


200 
235 
155 


97000 
104575 

83855 


EF 



420 

826 


60 
100 

75 


420 

406 


160 
175 


67200 
71050 



2) 1966029 

Area of part cut off by bases, 983014.5 
AGF 
AFD 
BCD 
DEF 

- 128 A., 2 E., 21.5 P. 



6295435 

4530917 

670475 

383567 
12863408.5 links. 



Sec. V.] SURVEYING FIELDS OF PARTICULAR FORMS. 



157 



The field-notes of a meadow, bounding on a river and 
divided into four fields, are as follows, — the measurements 
being to low-water mark. Eequired the map and the 
content of the whole: — 





D 




CM 

CO 

tH 


55 


1054 


72 


896 


97 


739 


75 


480 




C 



% 



255 



C 



CN 



CO 
CM 



1622 

1081 

B 




E 



CO 

CO 





B 




o 


63 


1414 


35 


1237 


87 


1016 


45 


824 


50 


652 




551 




452 




295 




D 





D 



CM 

Oi 

o 

CM 



1310 
992 
A 



Diagonal. 



toD 



C 



o 



1030 
A 



A 

CO 

1—1 

752 

E 



S71°E 



Area, 34 A., 3 R. 



To find the contents of the several enclosures, other lines 
would be required : these might be measured on the plat, 
if it were drawn with neatness and accuracy. 



158 CHAIN SURVEYING. [Chap. IV. 

SECTION VI. 

TIE-LINES. 

258. Tie-Lines. The external boundaries of a tract of 
land having more than three sides are not sufficient either 
for making a plat or calculating the area. In the methods 
heretofore laid down, diagonals were also used. In some 
cases, however, owing to obstructions, such as ponds, close 
woods, or buildings, it is difficult to run the diagonals. 
When this is the case, a line measured across one of the 
angles of a quadrilateral will determine the direction of two 
sides, and thus fix the relative position of all the lines of the 
tract. Such lines are called tie-lines. 

For example, suppose it pi g . 113. 

were required to survey the £ 

tract represented in Fig. 113, / -? c €V>?^ 
the interior of which is filled 1 wT-f^^JSM-M} 
with such thick woods that / ^ C^S'^^X 
the diagonals cannot be mea- / Md^'j^ ■'^uf'' 
sured : the external lines AB, 
BC, CD, and DA might be 
measured as before. Then / ..'''' 

on the lines adjacent to one / ,,.*-'' 

angle, as C, measure carefully f 

CE and CF ; also measure EF. These measures should be 
made with the greatest accuracy, as a slight error here will 
very materially affect the result. On the same account, the 
distances CE and CF should be taken as large as circum- 
stances will allow. 

If the tie-line cannot be run within the tract, the points 
may be taken at E and F in the sides produced. 

To plat such a tract, commence with the triangle. This 
being formed, the direction of CB and CD is known. 

259. To calculate the Area. First find in ECF the 
angle ECF, whence by trigonometry BD is found, and then 
the area of the triangles. 



Sec. VI.] TIE-LINES. 159 

If CE = CF, EF will be the chord of the arc to the 

EF 
radius CE, whence the chord to radius 1 = — — . This 

EC 

quotient being found in the table of chords the correspond- 
ing arc will give the degrees and minutes of the angle ECF : 
or CE : } EF : : rad. : sin. J ECF. 

260. Inaccessible Areas. By a combination of tie-lines 
and offsets, tracts that cannot be entered, such as a pond or 
a swamp, may be measured. For this purpose, surround 
the tract by a system of lines bound at the angles by tie- 
lines, and take offsets to the prominent points in the bound- 
ary of the tract. 

261. Defects of this Method. Every system of measure- 
ment or drafting should commence with the longer lines 
and end with the shorter. By this means the errors that 
are unavoidable are diminished as we proceed. If, for 
example, a diagonal of thirty chains were measured, this 
would fix the distance of the ends to a degree of certainty 
precisely equal to that of the measurement ; and if from this 
measurement the length of an inferior line joining two 
points in the sides were to be determined, the errors in 
the length of the diagonal would affect this length to a 
degree exactly proportional to its length, the error in a 
line of live chains long being one-sixth of that of the 
diagonal. Precisely the reverse is the case when the shorter 
line is measured : the error is magnified as we proceed. 
On this account, the method explained above should never 
be employed when it can be avoided. By the use of the 
compass, transit, or theodolite, this can always be done. 
The mode of using them for surveying purposes forms the 
subject of the next chapter. 



CHAPTER V. 

COMPASS SURVEYING. 



SECTION I. 

DEFINITIONS AND INSTRUMENTS. 

262. In chain surveying, the position of any point is 
determined either by directly measuring to it from other 
known points, or by determining its distance from such 
points by the indirect methods explained in last chapter. 
In the method about to be explained, its position is ascer- 
tained by angular measurements taken from known stations, 
or by its distance from a known point and the angle which 
it makes with the meridian. 

All those methods, which have a direct reference to the 
meridian as the base of angular distance, are known under 
the head of compass surveying; whether the instrument 
used to determine the angle is a theodolite, a transit, or a 
compass. 

263. The Meridian. If the heavens are examined during 
a clear night, the stars to the north will be perceived to 
revolve around a star elevated about 40°. This is called 
the pole-star, and is very nearly in the point in which the 
axis of the earth if produced would meet the heavens. 
This point is called the north pole of the heavens. The 
north star is not exactly at the pole, but revolves around it 
in a small circle. If a transit or theodolite be levelled, and 
the telescope directed to the centre of this circle (see 
chap, ix.) it will point exactly north. Depress it, and run 

160 



Sec. L] 



DEFINITIONS AND INSTRUMENTS. 



161 



out a line in the direction of the line of collimation. This 
will be a meridian line. 

264. The Points of the Compass. If through any 
station a line be drawn perpendicular to the meridian it will 
run east and west. If we face the south, the west will be 
to the right hand and the east to the left. These four points — 
north, east, south, west — are called the cardinal points of the 
compass, and are used as reference for all angular distances 
from the meridian. 

Fig. 114. 




For nautical purposes, each of the quadrants into which 
the horizon is divided is further divided into eight parts 
called points, and named as in Fig. 114, commencing at the 
north and going to the east. 

North, 1ST.; North by East, (N.5E.;) North Northeast, 
(N.N.E. ;) Northeast by North, (N.E.6N. ;) Northeast, (N.E. ;) 
Northeast by East, (N.E.6E. ;) East Northeast, (E.N.E. ;) East 
by North, (E.&N. ;) East, (E.) and so on, E.6S.; E.S.E.; 
S.E.6E.; S.E.; S.E.6S. ; S.S.E.; S.&E.; S. 

For land surveying only the cardinal points are men- 
tioned, the direction being determined by the angular dis- 
tance from the meridian. 



265. Bearing. The bearing of a line is the angle which 
it makes with a meridian through one end. It is expressed 
either by naming the points, as N.6E., S.S.E. J E., as is 

ll 



162 COMPASS SURVEYING. [Chap. V. 

done in navigation, or by mentioning the number of degrees 
in the angle accompanied by the cardinal points between 
which it runs. Thus, if a line runs between north and west 
and makes an angle of 37° 25' with the meridian, its bearing 
is K 37° 25' W. It deflects 37° 25' from the north towards 
the west, and is therefore sometimes said to run from north 
towards the west. This expression, though convenient, is 
not strictly correct. 

266. The Reverse Bearing. If the bearing of a line 
of moderate length is determined at one end, and then 
again at the other end, the latter is called the reverse bearing. 
It will be found to be of the same number of degrees as the 
bearing, but with the opposite points. Thus, if the bearing 
of a line be K 27J° E, its reverse bearing is S. 27|° W. 

If the line be long, there will be a continual variation 
from the initial course. Thus, if a line run 1ST. 45° E. through 
its whole course, it will be found to deviate to the left from 
a straight line. A true east and west line in latitude 40° 
is a curve with a radius of about 4800 miles. 

267. The Magnetic Needle. A magnetic needle is a 
light bar of magnetized steel suspended on a pivot, so that 
it may turn freely in a horizontal direction. Such a needle 
will always place itself in nearly the same direction, one 
end of it being northward and the other southward. The 
needle should move very freely on its pivot, so that it may 
always assume its proper position. The pivot should there- 
fore be of very hard steel ground to a fine point. In the 
centre of the needle there should likewise be a cup of agate 
or some other hard material inserted for it to rest upon. 

As the needle is generally balanced before being magnet- 
ized, the north end in northern latitudes will always "dip" 
after the magnetic force has been communicated to it. To 
restore the balance, a coil of fine brass wire is wrapped 
around the south end. This may be slipped along the bar 
so as perfectly to restore the balance. It serves also to dis- 
tinguish the two ends of the needle. 

A good needle will vibrate for a considerable time after 



Sec. L] DEFINITIONS AND INSTRUMENTS. 163 

having been disturbed. If it settles soon, it is defective in 
magnetic power, or the pivot is imperfect. To preserve the 
pivot in good order, the needle should always be lifted from 
it when not in use. 

268. The Magnetic Meridian. The line upon the sur- 
face of the earth in the direction of the needle, when unin- 
fluenced by disturbing causes, is called the magnetic me- 
ridian. If -the needle pointed steadily to the north pole, 
the magnetic meridian would coincide with the true. This 
is, however, far from being the case. Throughout the east- 
ern part of the United States and Canada it points west of 
north, the amount of the deviation (called the variation of the 
compass) being different in different places. This amount 
is subject to a gradual secular change. (See chap, x.) 

269. The Magnetic Bearing. The bearing of a line 
from the magnetic meridian is called the magnetic bearing. 
This has generally been used in land surveying. Its con- 
venience is such as to have heretofore counterbalanced its 
defects in the opinion of a large number of surveyors. The 
attention of scientific surveyors and legislators has of late 
been called to the difficulties arising from the use of such a 
false and varying standard. In Pennsylvania, by a late law, 
the bearings of all lines inserted in the title-deeds of real 
estate are required to be from the true meridian line. The 
surveys of United States public lands have always been 
made on this principle. 

270. There are two modes in which the needle may be 
employed to enable us to determine the bearing of a line. 

First. Attached to the needle may be fixed a card divided 
as in Fig. 114, or subdivided into degrees, — the north point 
of the needle being directly under the north point of the 
card. Such a card would always place itself in the same 
position with respect to the cardinal points. 

To determine the bearing of a line, it would only be 
necessary to have a pair of sights in the line of a diameter 
of the card, with an index between them to show at what 



164 COMPASS SURVEYING. [Chap. Y. 

point of the card the line crossed. The degrees between 
this point and the north or south point of the card would 
be the bearing required. Thus, the bearing of AB would 
be about !N". 67° E. The cardinal points on the card show 
the points between which the line runs. 

The great defect in this plan is that, in consequence of the 
weight of the card, the needle settles slowly, and the pivot 
is very liable to wear. The card, too, must be made of some 
light material, which cannot be divided so accurately as 
metal. This form is therefore never used except for the 
mariner's compass. 

Second. The sights may be connected with a circular box 
in the centre of which is the pivot, — the circumference of 
the box being appropriately divided. This is the plan em- 
ployed in the surveyor's compass or circumferentor. 

271. The Compass. The compass consists of a stiff 
brass plate A, (Figs. 115, 116,) carrying the circular box B, 
and furnished at the ends with two brass sights C, perpen- 
dicular to its plane. In the centre of the box is the pivot 
to support the magnetic needle. 

The circumference of the box is divided into 360°, and 
these in the larger instruments are subdivided into halves. 

The zero-points are in the line joining the sights, one 
being marked for the north, and the other for the south. 
The degrees are counted from zero to 90° each way. 

If we stand opposite the south point looking towards the 
north, the 90° on the left hand is marked E. and that on 
the right "W. The cardinal points thus follow each other 
in an inverted order. 

The reason why this should be so will appear from con- 
sidering the difference between the mariner's compass and 
the circumferentor. In the former, the card is stationary, 
while the index moves; in the latter, the index, which is the 
needle, is stationary, while the divided circle moves: while, 
then, the north point of the box is moving towards the east, 
the north point of the needle will traverse it towards the 
ivest. In order, then, that the index should not only point 
to the number of degrees, but also show the cardinal points 



Sec. L] 



DEFINITIONS AND INSTRUMENTS 



165 



between which the line runs, those points must be engraved 
in a reverse order. 

Thus, supposing the instrument to be in the position, (Fig. 
115,) the north point of the needle at L shows the magnetic 




north, and the south point the magnetic south; the point 
midway between these to the right is east. The line from C 
to C is therefore south of east. If then the north point of the 
needle is to be used as the index, it should be found between 
the letters S. and E. The bearing in the figure is S. 80° E. 



166 COMPASS SURVEYING. [Chap. V. 

272. The Sights. These consist of two plates of brass 
about an inch wide set at right angles to the plate. Each 
plate has a vertical slit cut in it, with larger openings at 
intervals, as seen in Fig. 116 at H. The faces of the sights 
are seen at G. The slits should be perfectly straight, and 
as narrow as is consistent with distinct vision. The largei 
openings enable the surveyor to see the object more readily 
than he could through the fine slits. 

Instead of the sights, a telescope that can be elevated or 
depressed in a plane perpendicular to that of the plate A is 
sometimes employed. It has the advantage of giving more 
distinct vision at great distances, and, when connected with 
a vertical arc, of determining the angle of elevation of a hill 
up or down which the line may run. This object may be 
obtained with the sights, by having at the lower end of one 
of them a projection pierced with a small hole, and upon 
the face of the other the angles of elevation engraved. By 
looking through the hole at an object on the summit of the 
hill, the angle of elevation may be read on the face of the 
engraved sight. 

If such a scale is not on the instrument, it may be put on 
by the surveyor himself; a mark being made on one sight 
near the bottom, or a small plate with a hole being screwed 
to it ; on the other, at the same distance from the plate, the 
zero mark should be made. The distance from zero to the 
other marks will be the tangent of the angle of elevation 
to a radius equal to the distance between the sights. 
Measure therefore accurately the distance between the 
sights, and say, As rad. : tangent of the number of degrees 
: : the distance between the sights : the distance from the 
zero point to the mark for that number of degrees. 

273. Attached to the plate there are generally two levels 
at right angles to each other, as in the transit and theodolite. 

274. The Verniers. In some instruments, the compass- 
box is movable about its centre for a few degrees, the 
amount of deflection being determined by the vernier V. 
The purpose of this arrangement will appear hereafter. 



Sec. L] DEFINITIONS AND INSTRUMENTS. 



167 



G 



=t 



(m 



s^> 



m» 



Fig. ne. 



II 



3 



*D 



N 



H" 



c 



0=0 



H 



=0 C= 



168 COMPASS SURVEYING. [Chap. V. 

275. In the figures 115, 116, the different parts described 
above are lettered as below. Different makers, however, 
arrange the parts differently. A is the principal plate, 
which bears all the other parts. B is the compass-box, 
sometimes movable about its centre by means of a pinion 
connected with the milled head I, and capable of being 
clamped in any position by the screw K. D is the needle, 
resting on a pivot in the middle of the compass-box. The 
needle can be raised from its pivot by the screw F. C and 
C are the sights, which are fastened to the plate by the 
screws !N\ M, M are the levels. 

276. The Pivot. This should, as remarked above, be 
extremely hard and very sharp. It should likewise be 
placed exactly in the centre of the box and in the line join- 
ing the slits in the sights. 

To discover whether it is properly centred, and likewise 
whether the needle is straight, turn the compass until the 
north point of the needle coincides with any given number 
of degrees. The south point must be 180° distant. If it 
is so in all positions, or, in four, distant 90°, as for instance 
the 0's and 90's, the needle is straight and well centred. 

Draw a hair or fine silk string through the slits in the 
sights. If this passes over the zero-points, the centre is in line. 

Or, sight to a very near object, and note the reading. Turn 
the instrument half round, and again note the reading : if 
these do not agree, the pivot is not on the line of sight. 
Half the difference is the actual error. 

277. The Divided Circle. The accuracy of the division 
may be tested by turning the plate into different positions. 
If in all cases the opposite ends of the needle point to the 
same number of degrees, the probability is that the circle is 
correctly divided. 

If the compass has a vernier, set the instrument in any 
direction. Then move the box through any number of 
degrees, and see whether the needle traverses the same 
number of degrees as the vernier. If it does in all posi- 
tions, the arc is properly divided. 



Sec. I.] DEFINITIONS AND INSTRUMENTS. 169 

278. Adjustments. The levels may be adjusted as 
directed for the transit and theodolite. 

The sights should be perpendicular to the plane of the 
instrument. To verify this, suspend a long plumb-line: 
level the plate, and sight to this line. If it appears equally 
distinct through all parts of the slit, the sight is perpen- 
dicular. Turn the instrument half round and test the other 
sight in the same manner. If either is found incorrect, the 
maker should rectify it. 

279. The compass, as already remarked, is very generally 
used for surveying purposes, though it is fast giving place 
to the transit. The latter is furnished with a compass-box, 
which was not described with the instrument, as it was not 
needed at that stage of the work. It is in all respects 
similar to the box attached to the compass itself. The 
theodolite likewise has a compass. It is, however, so small 
as to be of very little use in accurate work. 

280. The compass is generally supported on an axis in- 
serted in the socket 0. This axis terminates in a ball, 
which works freely but firmly in a socket. This arrange- 
ment admits of the axis being placed in any direction. 
The compass-plate may thus be made level. 

Instead of a tripod, many surveyors prefer a single staff 
pointed with iron. This is called a "Jacob's Staff." Its 
chief defects are the difficulty of setting in hard ground or 
among stones, and the want of steadiness in windy weather. 

281. Defects of the Compass. Though a very con- 
venient and useful instrument, the compass is deficient in 
two very important particulars: — its indications are neither 
correct nor precise. 

It is not correct, because, as already remarked, the needle 
(which is the standard) does not do what it professes : it 
does not point to the north. This would be of compara- 
tively little importance if its direction were fixed or paral- 
lel; but neither of these is the fact. It not only varies 



170 COMPASS SURVEYING. [Chap. V. 

from year to year, but from season to season, and even 
during the same day. These variations will be the subject 
of a future chapter. 

The presence of ferruginous matter in the earth, or the 
too great proximity of the chain, or of any other piece of 
iron, may deflect it very seriously from its normal position. 

It is not precise. The divisions on the arc are rarely 
smaller than half-degrees ; and if they were finer it would 
be difficult to read to less than a quarter of a degree. A 
little calculation will convince one that this is a serious 
defect where accuracy is desired. An error of 5' in the 
bearing would cause a deviation of nearly one foot in ten 
chains, or about seven feet eight inches in a mile. 



SECTION II. 
FIELD OPERATIONS. 

282. Bearings. To take the bearing of a line, set the 
compass directly over one end ; level it, and turn the plate 
till the other end of the line — or a rod set up in the direc- 
tion of the line at a distance as great as is consistent with 
distinct vision — can be seen through the slits. Then, when 
the needle has settled, notice the number of degrees to 
which the end of the needle points, and the cardinal points 
between which it is situated: the result will be the bearing 
of the line. 

If the north end of the compass is ahead, the north end 
of the needle should be used, and vice versa. 

If you are running with the north end of the compass 
ahead, and the north point of the needle is between S. and 
E. and points to 45J°, the bearing is S. 45J° E. 

In reading, the eye should be placed opposite to the other 



Sec. II. ] FIELD OPERATIONS. 171 

end of the needle; otherwise, owing to the parallax of the 
point, it will appear to stand at a different point of the arc 
from what it really does. Any iron about the person will 
be less likely to affect the needle than when in another 
position. 

283. Use of the Vernier. When the needle does not 
point to one of the divisions of the arc, it is usual to esti- 
mate the fraction. Some surveyors, however, after the 
needle has come to rest, notice between which divisions the 
needle points, and then move the compass-box, by turning 
the milled head I, until the point of the needle is op- 
posite one of the divisions. The amount by which the 
box is turned, as indicated by the vernier, will' give the 
fraction. 

This plan, though theoretically correct, adds really 
nothing to the correctness of the work. The liability to 
derangement, from handling the instrument, is so great as 
to neutralize any advantage it might otherwise possess. 

284. Reverse Bearing. The reverse bearing of every 
line should be taken. To do this, set the compass at the 
position of the rod, and sight back to the former station. 
The bearing found should be the reverse of the former. If 
it is not, the work at the former station should be reviewed ; 
if found correct, the difference between the two must arise 
from some local cause. 

285. Local Attraction. "When the back sight does not 
agree with the forward sight, some cause of derange- 
ment exists about one of the stations. This is called 
local attraction. It is generally caused by ferruginous 
matter in the earth. It is said that any high object, 
such as a building or even a tree, will slightly deflect the 
needle. In situations in which trap rocks abound, the 
local attraction is often very great. The author has known 
a variation of more than 10° in a line of two and a half 
chains long, produced by this cause alone. In such regions, 
running by the needle is very troublesome, and ma}^ cause 



172 COMPASS SURVEYING. [Chap. V. 

very serious errors unless great care is taken to allow for 
the effect produced. 

To discover where the attraction exists, select a number 
of positions in the neighborhood of the suspected points, 
and note their bearings from these stations, and also from 
each other. The agreement of several of these will prove 
their probable correctness. The points thus found to be 
void of local attraction may be taken as the starting 
points. 

In surveying a farm, a very good way is to note the 
forward and back sights of every line. If these are found 
to agree on any line, they may be presumed to be right, and 
the others corrected accordingly. 

286. To correct for back sights. 

When the back sight is greater than the fore sight, sub- 
tract the difference from the next bearing, if the two lie 
between the same points of the compass or between points 
directly opposite, but add it in all other cases. If the back 
sight is the less, add the difference in the former case, and 
subtract it in the latter. 

"Where the local attraction is great, or the line runs 
nearly in the direction of one of the cardinal points, a diffi- 
culty may occur in the application of the preceding rule. 
A little reflection will enable the surveyor to modify it to 
suit the case. 

287. By the Vernier. It is more convenient in practice 
to turn the box by the vernier until the reading for the 
back sight corresponds with the fore sight. The needle 
will then give the true bearing of the new line as though 
no attraction existed. 

288. To survey a Farm. Commence by goiug 
round it, and verifying, so far as can be done, the land- 
marks, fixing stakes at the corners, so that the assistant 
may readily find them if he is not already familiar with 
their position. Then, placing the compass at one corner, 



Seo. II.] FIELD OPERATIONS. 173 

send the flag-man ahead to the next corner ; note the bearing 
of his pole; and so proceed with the sides, in succession, 
taking a back sight at each station. 

If the end of the line cannot be seen from the bes;in- 
ning, let the flag-man erect his pole, in the line, at a point 
as distant from the beginning as possible. Sight to the 
pole, as before; then, going forward, set the compass by 
sighting to the last station. The flag-man should now be 
placed, exactly in line, at another station. So proceed 
until the end of the line has been reached. 

289. Random Line. If the first position of the flag- 
staff were not exactly in line, the course run will deviate 
to the right or left of the corner. Where such is' the case, 
measure the perpendicular distance to the corner, and de- 
termine the correction by the following rule : — 

As the length of the line is to the deviation found as 
above, so is 57.3 degrees, or 3438 minutes, to the correction 
in the bearing.* 

In running through woods, it is very frequently necessary 
to correct the bearing in this manner. In all cases, how- 
ever, where back sights are taken, the compass should be 
allowed to stand at the last station on the random line, 
since the local attraction often varies very considerably 
in a short distance. If it is desired to run the next line 
precisely on its location, the corner should be sighted to 
from the end of the random line, and a back sight 
taken. 



* This rule is founded on the ordinary rule for the solution of right-angled 
triangles, — the length being the hypothenuse, and the deviation the perpen- 
dicular, an arc of 57.3 degrees being equal in length to the radius. 

Thus, supposing, in running a line N. 35° 30 / E. 27.53 chains, the corner is 
found 35 links to the right hand : the calculation would be 

27.53 : 35 : : 57.3° : 0° 43'. 

The proper bearing would therefore be N. 36° 13' E. 



174 



COMPASS SURVEYING. 



[Chap. V. 



290. When the far end of the line cannot be seen, it 
will sometimes be found convenient to run to a station as 
near the middle of the line as possible, if one can be found 
from which both ends can be seen. Then, instead of con- 
tinuing on in the same course, sight to the corner. The 
chain-men should note the distance to the assumed station. 
A very obtuse-angled triangle will thus be formed, and the 
correction in bearing may be readily calculated. 

Thus, supposing the line were AB, (Fig. 117,) Fig. 117. 
passing over an elevation at C. At A the bearing 
of AC was found to be N. 43f° W., distance 
10.50 chains. At C, CB was K 43° W., distance 
7.36 chains. 

We have AC : BC :: sin. B : sin. A; 

or, as the angles are small, AC : BC :: B : A; 
whence AC + BC : BC : : B+A : A. 

That is, 17.86 : 7.36 :: 45' : A = 19', the required 
correction. The true bearing of AB is therefore 
K 43|° W. 

Where the deviation from the correct line is not much 
greater than in the example given, AB is sensibly equal to 
AC -f CB. Where the deviation is considerable, the angles 
and side should be calculated by Trigonometry. 

The above rule may be expressed thus : — 

As the sum of the distances is to the last distance, so is 
the whole deviation to the correction to be applied at the 
first station. 



291. Proof Bearings. In the course of the survey, 
bearings or angles should be taken to prominent objects. 
These form a test of the accuracy of the work. Three 
bearings are necessary to each object: two of these, being 
required to fix its position, will aflbrd no check on the inter- 
mediate measurements; but their coincidence with a third 
will determine the probable correctness of all, and of the 
connecting measurements. Diagonal bearings and dis- 
tances may likewise be taken as proof lines. 



Sec. II.] FIELD OPERATIONS. 175 

292. Angles of Deflection. In surveying with the 
transit or theodolite, it is most convenient to record the 
angles of deflection; that is, the angle by which the new 
course deviates to the right or to the left from that of the 
last line. This is always done in surveying roads, rivers, &c. 
From the angles of deflection the bearings are very readily 
deduced, by rules to be given hereafter. As checks to the 
work, the bearings of some of the lines may likewise be 
taken. 

In a closed survey the whole deflection must equal 360°. 
To determine whether it is so, arrange the deflections to 
the left in one column, and those to the right in another. 
Sum the numbers in each column : the difference of these 
sums should equal 360°. 

In practice this will rarely occur; though in open ground, 
where the angles can readily be taken, the error should not 
exceed four or five minutes in a tract of ten or twelve sides, 
provided a good transit or theodolite is employed. 

Example. 

The following are the notes of a survey taken by the 
author:— 1. S. 53° 10' W.; 2. Deflect 97° 3' to the right; 
3. 97° 45' to the right; 4. 81° 14' to the right; 5. 30° 
12' to the left; 6. 12° 14' to the left; 7. 27° 48' to the 



right. 


Whence the first line def 


Lects 98° 34' to 




Right hand. 


Left hand. 




97° 3' 


30° 12' 




97° 45' 


12° 14' 




81° 14' 


42° 26' 




27° 48' 






98° 34' 






402° 24' 






42° 26' 





359° 58', 
differing but two minutes from 360°. 



176 COMPASS SURVEYING. [Chap. V. 

Where the difference amounts to several minutes, it is 
best to distribute it among the angles. 

The rule which is sometimes given: to determine the 
angles from the bearings, and ascertain whether the sum 
of the internal angles is equal to twice as many right angles 
as the figure has sides, less four right angles — proves nothing 
in regard to the correctness of the field work. Any set of 
bearings will prove in this way. 



SECTION III. 

OBSTACLES IN COMPASS SURVEYING.* 

A.— PROBLEMS IN RUNNING LINES. 

293. Many of the obstacles that occur in angular sur- 
veying have already been alluded to. These, and all 
others which the operator will meet with, may be over- 
come by the principles of Trigonometry. As, however, 
there is frequently a choice in the means to be used, the 
following methods are given, as being perhaps the most 
simple : — 

294. Problem 1. — To run a line making a given angle with 
a given line from a given point within it 

Place the instrument at the point, and sight along the 
line. Tarn the plate the required number of degrees, and 
the sights or telescope will be in the required line. 

>; Many more such methods may be found in Gillespie's "Land Surveying." 



Sec. III.] 



OBSTACLES IN COMPASS SURVEYING. 



177 



295. Problem 2. — To run a line making a given angle with 
a given inaccessible line at a given point in that line. 

Let AB (Fig. 118) be the given rig. us. 

line, and A the given point. 
Take two points C and D from 
which A and some other point B 
in AB may be seen, and measure 
CD. Then take the angles ACD, 
BCD, ADC, and BDC. The dis- 
tance AC and the angle CAB 
may be calculated. 

Run CE, making ACE = CAB : CE will then be parallel 
to AB. Now, if we suppose AE to be drawn, ' we shall 
have in the triangle ACE all the angles and side AC to 
find CE. Lay off this distance from C to E, and run the 
line EF towards A. 

If A cannot be seen from E, calculate CEF, and run the 
line from E, making the proper angle with CE. 




Problem 3. — From a given point out of a line, to run a 
line making a given angle with that line, 

296. Where the line is accessible. 

If the compass is used. Take the bearing of the given 
line. Then place the compass at the given point, and 
set it to same bearing. Deflect the compass the number of 
degrees required, and run the line. 

If a transit or theodolite is used. Fi g- 119 « 

Set the instrument at some point 
A (Fig. 119) in the line, and take 
the angle BAC. Move the instru- \ 

ment to C, and make the angle 
ACB = B - A, or = 180° - (B + 
A), and CB or CB' will be the line required. 

In all cases, unless the line is to be a perpendicular, there 
will be two lines that will answer the conditions. 

12 




178 



COMPASS SURVEYING. 



[Chap. V. 




297. If the line is inaccessible. Let m %- 12 °- 

A F 

AB (Fig. 120) be the given line, and — *r *; — 

C the given point. Run any con- 
venient base CD, and take the angles 
of position of two visible points A 
and B in the given line. Then, in the 
triangle ADC, we shall have DC and 
the angles, to find CA. Similarly, in CBD, find CB. 
Then, in ACB, we shall have AC, CB, and ACB to find 
ABC. 

Run CF, making BCF = B - F, or 180° - (B +F), and 
it will make the required angle with AB. 

298. If the point be inaccessible. 
From any convenient stations A 
and B (Fig. 121) in the line AB, 
take the angles of position of the 
point C, and measure AB. Then, 
in the triangle ABC, we shall 
have the angles and the side AB 
to find BC. 

In BCD we then have the angles and side BC to find 
BD. 

BD may be found by a single proportion, thus : — 

Sin. ACB . sin. BDC : sin. BAC . sin. BCD : : AB 




BD. 



For we have sin. ACB : sin. BAC : : AB : BC, 
and sin. BDC : sin. BCD : : BC : BD. 

Whence (23.6) 
sin. ACB . sin. BDC : sin. BAC . sin. BCD : : AB 



BD. 



Having found BD, DC may be run towards C; or by the 
angle, if C be invisible from D. 

If C is visible from the point D, the latter may be found 
by trial, thus : — 

Set the instrument at a station as near the proper posi- 
tion as possible, and deflect the given angle. Notice 
whether the line passes to the right or left of the point, and 



Sec. III.] 



OBSTACLES IN COMPASS SURVEYING. 



179 



move the instrument accordingly. A few trials will put it 
in its proper place. 



Fig. 122. 
G 




299. If the point and the line both 
be inaccessible. Take any convenient 
station D, (Fig. 122,) and run DE 
parallel to AB, by Art. 302. Then 
run CFG, making the required angle 
with ED, by Art. 298 ; or the dis- 
tance on the base DC (Fig. 125) 
may be calculated. 



Problem 4. — To run a line parallel to a given line through a 
given point. 

300. If the line he accessible. 

With the compass. Take the bearing of the given line, and 
through the given point run a line with 
the same bearing. 

With the transit or theodolite, At any 
point A (Fig. 123) in the given line 
take the angle BAC. Eemove the 
instrument to C, and make ACD = d " 

BAC. CD will be parallel to AB. 

301. If the point be inaccessible. Fi s- 124. 
At A and B, (Fig. 124,) any two 
points in the given line, take the 
angles BAC and ABC. Measure 
AB, and calculate AC. Make CBD 
= ACBandBD=AC. Through 
D run DE in the line CD : it will be the parallel required. 



Fig. 123. 



A 




302. If the line be inaccessible. 
From C (Fig. 125) run any base- 
line CD ; and at C and D take 
the angles of position of two 
visible points A and B in the 
given line. Calculate the angle 




180 



COMPASS SURVEYING. 



[Chap. V. 



CAB. Bun EOF, making ACE = CAB, and EF is the 
parallel required. 



If the line and the 'point both be inaccessible. 



Fig. 126. 




Fig. 127. 



303. First Method.— Assume 
any station D, (Fig 126,) and 
run a line DE parallel to AB, 
by Art. 302, and towards C run 
Fa parallel to DE, by Art. 301. 



304. Second Method. — 
Take any convenient base 
DE, (Fig. 127,) and take 
the angles of position of 
C, A, and B at D and E. 
Calculate BE, CE, and 
EBA. Then CFB = 180° 
- EBA. In CEF, we 
then have the angles and 
CE to find EF. Lay off EF the calculated distance, and run 
the line from F to C. 




B — PROBLEMS FOR THE PROLONGATION AND INTER- 
POLATION OF LINES. 

305. In running a line, obstacles are often met with 
which it requires some ingenuity to overcome, and which 
will perplex the surveyor unless he has prepared himself 
by previous study of all cases which are likely to occur. 
If the total length of a line were all that it was necessary 
to determine, the two points at its extremity might be con- 
nected by a series of triangles, and that length calculated 
by Trigonometry; but it is generally desirable to have the 
line marked out so that the exact position of the dividing 
fence, if one is placed, or of the division if there be no 
fence, may be indicated by stakes or by marked trees. 
To do this, the line itself must be traced, or another run 



Sec. Ill] 



OBSTACLES IN COMPASS SURVEYING. 



181 



in its neighborhood, so related to that in question that the 
surveyor can at any time pass from the one to the other to 
set his landmarks. "We shall treat of the different kinds 
of obstructions likely to occur; and, as the prolongation 
and interpolation of the lines are generally closely con- 
nected with the determination of their lengths, the two will 
be considered together. 

Problem 1. — To prolong a line beyond a building or other 
obstruction. 

306. First Method. — At a point of the line erect a per- 
pendicular of such length as to pass beyond the obstacle. 
Through the extremity of this run a parallel to the given 
line : after passing the obstacle, pass back to the required 
line by an equal perpendicular. The distance will be equal 
to that of the parallel. 



307. Second Method. — At B 
(Fig. 128) deflect 60°, and mea- 
sure BC. At C deflect 120°, 
and measure CD = BC. Deflect 
60°, and run DE, which will be 
in line with AB. BD = BC ; for 
BDC is an equilateral triangle. 



Pig. 128. 




308. Third Method.— At 
B (Fig. 129) deflect 60°, 
and measure BC. At C 
deflect 90°, and measure 
CD =1.732 times BC. At 
D deflect 30°, and DE will 
be in line with AB. BD = 2BC. 



Pis:. 129. 




309. Fourth Method.— At 
B (Fig. 130) deflect 45°. A 
Measure BC. At C turn 
90°, and make CD = BC. 



Pig. 130. 




At D turn 45°, and DE will be in line. BD = 1.414 BC. 



182 COMPASS SURVEYING. [Chap. V. 

Problem 2. — To interpolate points in a line. 

310. If one end be visible from the other. Set the instru- 
ment at one end and sight to the other: an assistant can 
then be signalled to place stakes directly in line. In 
crossing a valley, determine a station, as above, on the 
borders, from which the valley can be seen; and, placing 
the instrument at this point, sight to a similarly deter- 
mined station on the other side. Stations may thus be 
determined down a very considerable declivity. With the 
transit almost any slope may be sighted down. In this 
operation, the instrument must be very carefully levelled 
sideways ; otherwise, the points determined in the valley 
will be out of line. 

311. By a Random line. If a wood, or other ob- Fi s- 131 « 



A 



struction, prevents one end of the line, as B, (Fig. 
131,) from being seen, run a line AC as nearly in 
the given course as possible, and drive a stake every 
five or ten chains, or oftener if desirable. "When 
you have arrived opposite the end of the line, note 
the distance. Also measure the distance CB to 
the end. The correction of the bearing may be 
found as in Art. 289, and the points be inter- 
polated as in Art. 209. 

312. If the line cannot be run from the first station. 

Lay off AC (Fig. 132) as nearly perpendicular Fig.^132. 
to the line as possible, and run the random line 
CD. On arriving opposite the end, measure DB. 
Then say, — 

As CD is to the difference between BD and AC, 

so is 57.3°, or 3438', to the correction of J_j 

bearing. 



c 



i ; 
1 1 

/ I 

( I 

i I 

.' I 



/ 



Li 



To interpolate points — Say, as CD is to the 
distance Ca to any station on the random line, so 
is the difference between BD and AC to a fourth d 1 -- ~--k 



\a 



\ 
\ 
\ 

Ah 
\ 

\ 
\ 



- V D 



Sec. III.] OBSTACLES IN COMPASS SURVEYING. 18o 

term. This fourth term added to AC if BD is greater than 
AC, but subtracted if it be less, will give the correction for 
the point a. 

If the random line crosses the other, as in Fig. Fi s- 133 - 

G A 

133, say, As CD is to the sum of AC and BD, so 
is 57.3°, or 3438', to the correction of the 
bearing. 

Points may be interpolated by the following 
rule : — 

Say, As CD is to the sum of AC and BD, so 
is the distance Ca to any point in the random line 
to a fourth term. Take the difference between 
this fourth term and AC. c \~d' 

Then if AC is the greater of the two, lay off j 
the difference on the same side of the random e'"~~_ 
line that A is; but if AC be the less, lay off the remainder 
on the opposite side. 

Where a point in the line at a given distance from the 
beginning is required, measure that distance on the ran- 
dom line, and determine the offset as above. 

If the random line comes out very distant from the far 
station, it is better to run another than to depend on that 
as a basis for interpolation. 

C.— PROBLEMS FOR THE MEASUREMENT OF INAC- 
CESSIBLE DISTANCES. 

313. The various methods of determining the lengths of 
inaccessible lines are merely applications of the rules of 
Trigonometry, and might, therefore, be applied by the stu- 
dent without further instruction. There is, however, 
always a choice in the method to be employed: the fol 
lowing are therefore given, that all that is needful in the 
case may be brought together. 

Problem 1. — To determine the distance between two points 
which are accessible and visible from each other. 



184 



COMPASS SURVEYING. 



[Chap. V. 



Fig. 134. 




Fig. 135. 



314. First Method. — Select any 
station C, (Fig. 134.) Measure BC, 
and take the angles BAC and b 
ABC. Thence we can calculate AB. 

315. Second Method. — Measure 
CA and CB (Fig. 134) and the 
angle ACB; whence, having two 
sides and the included angle, AB 
may be determined. 

316. Third Method.— Where the 
angles can be taken to the ex- 
tremities of an inaccessible but 
known base CD, (Fig. 135,) the dis- 
tance AB may be calculated 
thus : — 



In ABD we have AD : AB : : sin. ABD : sin. ADB, 
and in ABC we have AB : AC : : sin. ACB : sin. ABC. 

Whence (23.6) AD : AC : : sin. ABD . sin . ACB : sin. ADB . 
sin. ABC. 

Then, in CAD having the ratio of AC to AD and the 
angle CAD, we may find the other angles by Art. 141, 
thus : — 

As AD : AC, or sin. ABD . sin. ACB : sin. ADB . sin. 
ABC : : r : tan. x, and as rad. : tan. (^^45°) : : tan. j- (ACD 
+ ADC) : tan. J (ACD *» ADC.) 

Having now the angles and one side of ACD, AD is 
found ; whence, in ADB, AB may be determined. 




Thus, 
and 



sin. CAD : sin. ACD : : CD : AD, 
sin. ABD : sin. ADB : : AD : AB. 



Whence (23.6) sin. CAD . sin. ABD : sin. ACD. sin. ADB 
: CD : AB. 



Sec, III.] 



OBSTACLES IN COMPASS SURVEYING. 



185 



Examples. 

To determine the distance AB, accessible at its extremi- 
ties, I took the angles to the ends of a line CD 10.75 
chains long, as follows:— B AC = 100° 35'; BAD, 48° 19'; 
ABC, 46° 15'; and ABD, 85° 23'. Required the distance 
AB. 

ACB = 180° - (BAC + ABC) = 33° 10'. 
ADB = 180° - (BAD + ABD) = 46° 18'. 



A _ r sin. ABD 
AsADor isin.ACB 


85° 23' A. 


C. 0.001411 


33° 10' " 


" 0.261952 


, _. f sin. ADB 
: AC or | sim ABC 


46° 18' 
46° 15' 


9.859119 

9.858756 


: : rad. 




10.000000 


: tan. x 


43° 45' 46" 
45 


9.981238 


tan. 45° — x 


1° 14' 14" 


8.334392 


ACD + ADC 
tan. 

2 


63° 52' 


10.309258 


ACD - ADC 

tan. 

2 


2° 31' 14" 


8.643650 


ACD 


66° 23' 14" 




k f sin. CAD 

Then, As < . A x^ 
' t sin. ABD 


52° 16' A. 


C. 0.101896 


85° 23' " 


" 0.001411 


( sin. ACD 
\ sin. ADB 


66° 23' 14" 


9.962025 


46° 18' 


9.859119 


:: CD 


10.75 ch. 


1.031408 


: AB 


9.034 ch. 


0.955859 



Problem 2. — To determine the distance on a line to the in- 
accessible but visible extremity. 

317. This may be done by the methods explained in 
Arts. 236, 237, and 238, using the transit or theodolite in 
running the lines, or by the following method : — 

318. Run a base line from a point in the line making any 



186 



COMPASS SURVEYING. 



[Chap. V. 



angle therewith, and at its extremity take the angle of posi- 
tion of the point. A triangle is thus formed of which the 
angles and one side are known. 

In this operation the triangle should be made as nearly 
equilateral as possible. 



Problem 3. — To determine the distance when the end is in- 
visible and inaccessible. 



319. First Method.— De- Vig.ue. 
fleet at B (Fig. 136) by any 
angle, and measure BD to a 
point from which C is visi- 
ble. TakeBDC. Thencalcu- A 

late BC. The angle C should 
be made as large as possible. 

If AB will not certainly 
pass through C, operate by the second method. 







¥&um 



320. Second Method. — Run 
EBD making any angle with 
AB, (Fig. 137.) Take the angles 
D and E. In DEC find DC. 
Then in DCB we have two sides 
DC and DB and the included 
angle to find BC and DBC. If 
DBC is equal to ABE, C is in 
AB produced. 



Fig. 137. 






-vc 



Problem 4. — To determine the distance to the intersection of 
two inaccessible lines. 



Sec. III.] 



OBSTACLES IN COMPASS SURVEYING. 



187 



321. Let AB and 

CD (Fig. 138) be the 
lines, their intersec- 
tion E being both in- 
visible and inaccessi- 
ble. It is required to 
run a line from a 
given point G, that 
shall pass through E, 
and to determine GE. 

Run any base line 
GH, and take the angles of position of the points A, B, C, 
and D on the given lines. 

Find GC, GD, and GDC ; also GA, GB, and GBA. Then, 
in GBD, we have GB, GD, and BGD, to find GBD, GDB, 
and BD. In BDE we then have BD and the angles to find 
BE. Finally, in GBE we have GB, BE, and the included 
angle, to find BGE and GE. 




If the lines AB and CD 

were accessible, the line GE 

might be run by Art. 212, 

and the distance determined 

by taking the angles C and 

G, (Fig. 139.) 

sin. GCE ^ 
Then GE = — — - - GC. 
sin. GEC 



Fig. 139. 




~:^--"e 



Problem 5. — To determine the distance between two inac- 
cessible points. 



322. First Method.—. .Select if . Fi s- uo - 

possible a point C, in the direc- ^ 1 

tion of the line AB, (Fig. 140.) N >» % 

From a station D, take ADB v *^ 

and BDC, and measure DC. 

Then in CDB we have CD 

and the angles to find CB, and 

in CDA we have CD and the angles to find CA. 

AB = CA - CB. 



D 



188 COMPASS SURVEYING. [Chap. V. 

323. Second Method.— Take a base line CD, (Fig. 135,) 
which, if possible, should be chosen nearly parallel to AB, 
and not much shorter than it. From C and D take the 
angles of position of A and B, whence AB may be calcu- 
lated. 

324. Third Method. — If no two points can be found 
whence A and B can both be seen, the distance can be found 
as in Prob. 9, p. 114. 

325. Fourth Method. — If A and B can both be seen from 
no one station, the distance may be found by Prob. 13, 
p. 116. 

326. Examples illustrative of the preceding rules. 

Ex. 1. It being necessary to run a parallel to a given in- 
accessible line AB, so as to pass through a given point C, 
also inaccessible and probably invisible from any point in 
the proposed line, I took a base line DE (Fig. 127) of 18 
chains, and at D and E determined the following angles of 
position,— viz. : EDO = 106° 35'; EDA = 72° 5'; EDB = 
21° 20'; DEC = 26° 50'; DEA = 61° 20'; and DEB = 120° 
45'. Required the distance DG and the angle DGF ; also 
the distance GC to the given station. 

Ans. DG 8.48 ch., GC 13.47 ch., and DGF = 124° 8' 17". 

Ex. 2. One side AB of a tract of land being inaccessible, 
and it being required to run from a given station C a line 
which shall make an angle of 67° 35' with that side, I 
measured a base line CD of 7 chains, and took the angles 
CDA = 100° 25'; CDB = 47° 29'; DCA = 32° 17'; and 
DCB = 90° 3'. Required the angle DCF which the required 
line makes with DC ; also the distance on CF to the line 
AB, and the distance of the point of intersection from A. 
Ans. DCF = 49° 10' 20", CF = 7.84, AF = 2.94. 

Ex. 3. The line AB not being accessible except at its ex- 
tremities, which were, however, visible from each other, I 
took the angles as follow to the points C and D, whose dis- 
tance I had previously found to be 10.78 chains, and found 



Seg. III.] OBSTACLES IN COMPASS SURVEYING. 189 

them to be BAD = 46° 30'; BAC = 81° 43'; ABC = 37° 
23'; and ABD = 80° 47'. Required AB. 

Ans. AB = 13.76 ch. 

Ex. 4. To a given inaccessible line AB it being required 
to run a perpendicular which shall pass through a point P 
also inaccessible, I took a base CD of 15 chains, and mea- 
sured the angles as follow,— viz. : DCP = 105° 30'; DCA 
= 256° 50'; DCB = 326° 42'; PDC = 38° 50'; PDA = 
79° 38'; PDB = 131° 7'. Eequired the distance on DC 

from D to the proposed line. 

Ans. DF = 14.36. 

Ex. 5. One side AB of a tract of land being inaccessible, 
and it being required to locate the adjoining side AE, which 
makes with the former an angle BAE of 98° 17', a base CD 
of 10 chains was measured. At C, the angle DCA was 95° 
and DCB = 37° 20'. At D, CDA was 43° 45', and CDB 
= 87° 39'. Required the angle between CD and a parallel 
to AB ; also the distance on that parallel to the point E in 
AE, and the distance AE. 

Ans. The parallel makes with CD the angle DCE = 163° 
57', CE = 5.19 ch., and AE = 9.89 ch. 

Ex. 6. In running a random line AB N". 87° E. towards a 
point C, after proceeding 7.50 chains I came to an impass- 
able swamp. I therefore measured on a perpendicular 
K 3° W. 4.25 chains, and S. 3° E. 5 chains to the points D 
and E from which C could be seen. At D, the angle CDE 
was 66° 39', and at E, DEC was 67° 25'. Required the dis- 
tance BC, the true course and distance of AC. 

Ans. BC = 10.93 ch. ; AC = 18.42 ch. ; True course 
K 88° 26' E. 



190 



COMPASS SURVEYING. 



[Okap. V. 



SECTION IV. 

FIELD-NOTES. 

327. The field-notes, when the bearings are taken, are 

recorded in various modes. 

First Method. — The simplest method is to write them 
after each other, as ordinary writing, thus : — 

Beginning at a limestone corner of James Brown's land, 
"N. 27J° E. 7.75 chains, to a marked white-oak. Thence, 
S. 60J° E. 10.80 chains, to a limestone, &c. 

In recording the boundaries, it is well to name the pro- 
prietors of the adjoining properties. These are always 
inserted in deeds of conveyance. 

328. Second Method. — Rule three columns, aa in the ad- 
joining plan: in the first, insert the station; in the 
second, the bearing; and, in the third, the distance: the 
margin to the right will serve for the landmarks, adjoining 
proprietors, &c. The left-hand page of the book may be 
reserved — as directed in Chain Surveying — for remarks, 
subsidiary calculations, &c. 



Sta. 
1 

2 
3 
4 
5 
6 


Bearing. 

N.27|°E. 
S.62|°E. 
S. 80° E. 
S.47i°E. 
S. 54| W. 
K37J°W. 


Distance. 


Landmai-ks, &c. 


7.75 
10.80 
9.50 
9.37 
8.42 
23.69 


to a marked white-oak. 
" limestone. 
" do. 

" forked white-oak. 
u limestone. 
" do. the place of beginning. 



329. Third Method. — Where there are subsidiary mea- 
surements, — such as offsets, intermediate distances, &c, — 
the above method is not convenient, as it requires a new 
table for each line along which such measurements are 



Sec. IV.] 



FIELD-NOTES. 



191 



made. In such, cases, the method by columns, with mar- 
ginal sketches of fences, streams, &c, is perhaps the best. 
The notation for " False Stations," the crossing of lines, 
streams, &c, (adopted in Art. 244,) may be employed here. 
The bearing should be inserted diagonally in the columns, 
and the bearings of cross fences, proof bearings, with the 
offsets, should be recorded in the right or left-hand margin, 
according as the lines or points to which they refer are to 
the right or left of the line being run. 

Sketches of the adjoining fences may likewise be inserted 
in the margin, with the distances to the intersections. By 
this combination of the columns and sketches, all the field- 
work may be recorded concisely, luminously, and accu- 
rately. 

The following notes of a survey will illustrate the 
above : — 



P5 






e3 

a 

© 

H 

o 







Sta. 4 




B 





1132 






55 


1054 




CO 


72 


896 




o 


97 


739 






75 


480. 




to 




s, nn * 




o 




Sta. 3 








Sta. 3 


Limestone on 






1450 


bank of run 






1030 


<*°, 








^^ 






Sta. 2 








Sta. 2 


a limestone. 






1344 










N. 59°10'E. 










Sta. 1 


a limestone. 




d 
o 





Sta. 5 




1740 


63 


1414 


35 


1237 


87 


1016 


45 


824 


50 


652 




551 




452 




295 




^.tf* 




Sta. 4 



a marked tree, 
corner of Wm. 
Phillips's. 




192 



COMPASS SURVEYING. 



[Chap. V. 



Fig. 141 is a plat of this tract. 

Fig. 141. 




SECTION V. 

LATITUDES AND DEPARTURES, 

DEFINITIONS. 

330. The difference of latitude — or, as it is concisely called, 
the latitude of a line — is the distance one end is farther 
north or south than the other. 

It is reckoned north or south according as the bearing is 
northerly or southerly. 

331. The difference of longitude or the departure of a line is 
the distance one end is farther east or west than the other, 
and is reckoned east or west as the bearing is easterly or 
westerly. * 

332. Where the course is directly north or south, the 
latitude is equal to the distance, and the departure is zero ; 
hut where the bearing is east or west, the latitude is zero, 



Sec. V.] LATITUDES AXD DEPARTURES. 193 

and the departure is equal to the distance. In all other 
cases the latitude and departure will each be less than the 
distance, the latter being the hypothenuse of a right-angled 
triangle, of which the others are the legs, and the angle 
adjacent to the latitude the bearing. Thus, T U2 

AB (Fig. 142) being the line, AC is the N ; 
latitude north, and CB the departure east. 

Strictly speaking, the triangle is a right- 
angled spherical triangle ; but the deviation 
from a plane is so small as to be abso- 
lutely unappreciable except in lines of 
great length. !N"o notice is, therefore, taken 
of the rotundity of the earth in "Land i 
Surveying." i 

333. The latitude, departure, and distance being the sides of 
a right-angled triangle, of which the bearing is one of the acute 
angles, any two of these mag be found if the others are known. 

1. Given the bearing and distance, to find latitude and 
departure. 

As radius : cosine of bearing : : distance : latitude ; 
and as radius : sine of bearing : : distance : departure. 

2. Given the latitude and departure, to find the bearing 
and distance. 

As latitude : departure : : radius : tangent of bearing. 
As cosine of bearing : radius : : latitude : distance. 

3. Given the bearing and departure, to find the distance 
and latitude. 

As sine of bearing : radius : : departure : distance. 

As radius : cotangent of bearing : : departure : latitude. 

4. Given the bearing and latitude, to find the distance 
and departure. 

As cosine of bearing : radius : : latitude : distance. 
As radius : tangent of bearing : : latitude : departure. 

13 



194 COMPASS SURVEYING. [Chap. V 

5. Given the distance and latitude, to find the bearing 
and departure. 

As distance : latitude : : radius : cosine of bearing. 
As radius : sine of bearing : : distance : departure. 

6. Given the distance and departure, to find the bearing 
and latitude. 

As distance : departure : : radius : sine of bearing. 
As radius : cosine of bearing : : distance : latitude. 

Examples. 

Ex. 1. Giving the bearing and distance of a line !N". 56^° 
\V. 37.56 chains, to find the latitude and departure. 

Ans. Lat. 20.87 K; Dep. 31.23 W. 

Ex. 2. Given the difference of latitude 36.17 !N\, and the 
distance 52.95, to find the bearing and departure, east. 

Ans. Bearing = K 46° 55' E.; Dep. = 38.67. 

Ex. 3. Given the difference of latitude 19.25 N"., and the 
departure 26.45 "W"., to find the bearing and distance. 

Ans. Bearing = K 53° 57' W. ; dist. = 32.71. 

Ex. 4. Given the bearing S. 33J° W., and the departure 
18.33 chains, to find the distance and difference of latitude. 
Ans. Dist. = 33.21 ch.; Lat. = 27.69 S. 

334. Traverse Table. The traverse table contains the 
latitudes and departures for every quarter degree of the 
quadrant to all distances up to ten. Erom these, the lati- 
tude and departure, corresponding to any bearing and dis- 
tance, may readily be found by the following rule : — 

If the distance be not-greater than ten. — Seek the degrees at 
the top or bottom of the table according as their number is 
less or greater than 45°, and in the columns marked Lati- 
tude and Departure, opposite to the distance, will be found 
the latitude and departure. If the degrees are found at the 
bottom of the table, the name of the column is there like- 
wise. Eor all degrees less than forty five, the left-hand 



Sec. V.] LATITUDES AND DEPARTURES. 195 

column is the latitude, but the departure, for those greater 
than 45°. 

If the distance be more than ten, and consist of whole tens. — 
Take out the number from the table as before, and remove 
the decimal point as many places to the right as there are 
ciphers at the right of the distance in the table. 

If the distance is not composed simply of tens. — Take from 
the table the latitude and departure corresponding to every 
figure, removing the decimal point as many places to the 
right or to the left as the digit is removed to the left or the 
right of the unit's place, and take the sum of the results. 

Examples. 
Ex. 1. Required the latitude and departure of a line 
bearing K 37|° E. 8 chains. 

Opposite to 8 chains, under the degrees 37J, are found, — 

Lat. 6.3680, Dep. 4.8424. 
The latitude and departure required are, therefore, 

6.37 K, 4.84 E. 
If the distance had been 80 chains, the latitude and de- 
parture would have been 

63.68 K, 48.42 E. 

Ex. 2. Required the latitude and departure of a line run- 
ning S. 63J° E. 75 chains. 

70 ch. Lat. 31.234 D ep . 62.645 

5 " 2.231 4.475 

33.465 67120 

Hence the result is Lat. 33.46 S. ; Dep. 67.12 E. 

Ex. 3. Required the latitude and departure of a line run- 
ning K 35}° W. 58.65 chains. 

50 ch. Lat. 40.579 Dep. 29.212 

8 " 6.493 4.674 

.6 487 351 

.05 41_ 29 

Lat. 47.600 K Dep. 34.266 W. 



196 COMPASS SURVEYING. [Chap. V. 

Ex. 4. "What are the latitude and departure of a line bear- 
ing S. 63|° W. 27.49 chains ? 

Ans. Lat. 12.27 S.; Dep. 24.60 W. 

Ex. 5. What are the latitude and departure of a line "N". 55 j- ° 
E. 27 chains ? Ans. Lat. 15.20 1ST.; Dep. 22.32 E. 

Ex. 6. What are the latitude and departure of a line bear- 
ing K 84f° E. 123.56 chains? 

Ans. Lat. 11.31 K; Dep. 123.04 E. 

Ex. 7. What are the latitude and departure, the bearing 

24f° W. 97.56 chains? 
Ans. Lat. 88.60 S. ; Dep. 40.84 W. 



and distance being S. 24f ° W. 97.56 chains ? 



335. When the bearing is given to minutes. Take out the 
numbers in the table for the quarter degrees between which 
the minutes fall. Then say, — 

As 15 minutes is to the excess of the given number of 
minutes above the less of the two quarters, so is the dif- 
ference of the numbers in the table to a fourth term, which 
must be subtracted from the number corresponding to the 
less of the two quarters if the quantity is a latitude, but 
added if it is a departure. 

Thus, supposing the line were "N. 41° 18' E. 43.27 chains. 
Take the difference between the latitude for 41 J° and that 
for 41J\ and say, — 

As 15' is to the difference between 41J° and 41° 18', or 3', 
so is the difference between the latitudes to the correction 
for 3'. This correction subtracted from the latitude for 
41J° will give the latitude required. 

Do the same with the departure, except that the correc- 
tion found as above must be added to the departure for 41 J°. 

In the example, we have for the distance 40 in the 
column for 

41i° the Lat. 30.074 Dep. 26.374 

41J° 29.958 26.505 



Differences .116 .131 

Then, As 15' : 3' : : .116 : .023, correction of latitude ; 
and, As 15' : 3' : : .131 : .026, correction of departure. 



Sec. V.] LATITUDES AND DEPARTURES. 197 

The corrected latitude and departure for 41° 18', distance 
40 chains, are Lat. 30.051., Dep. 26.400. 

In like manner, the latitudes and departures for each of 
the remaining figures may be calculated, being as below : — 

For 40 ch. Lat. 30.051 Dep. 26.400 

3 " 2.254 1.980 

.2 150 132 

.07 53 46 



32.508 K 28.558 E. 

There will rarely be any calculation necessary for the 
decimal figures of the distance, as the variation caused by 
a quarter of a degree will seldom change more than a unit 
any of the figures that need be retained. 

Ex. 1. The bearing and distance being "K. 76° 42' E. 39.76 
chains, to find the difference of latitude and departure. 

Ans. Lat. 9.147 K ; Dep. 38.694 E. 

Ex. 2. Given the bearing and distance S. 37° 9' E. 63.45 
chains, to find the difference of latitude and departure. 

Ans. Lat. 50.573 S.; Dep. 38.317 E. 

Ex. 3. Required the difference of latitude and departure 
L 29° 17' E. 123.75 chains. 
Ans. Lat. 107.937 S. ; Dep. 60.529 E. 



of a line running S. 29° 17' E. 123.75 chains. 



336. By Table of Natural Sines and Cosines. The differ- 
ence of latitude and departure, when the bearing is given 
to minutes, is more readily found from the table of natural 
sines and cosines than from the traverse table. The dif- 
ference of latitude and departure are the cosine and the 
sine of the bearing to a radius equal to the distance. 
Therefore, to find the difference of latitude and departure 
of a line, take out the natural cosine and sine of the bear- 
ing, and multiply them by the distance. 

Ex. 1. Required the difference of latitude and departure 
of a line bearing X. 41° 18' E. 43.27 chains. 



198 COMPASS SURVEYING. [Chap.V. 



41° 18' 


C( 


)sine .75126 


Si 


ne 66000 


Dist. 


DiflF. Lat. 


Dep. 


40 ch. 




30.0504 




26.4000 


3 « 




2.2538 




1.9800 


.2 




1503 




1320 


.07 




526 




462 



Lat. 32.5071 N. Dep. 28.5582 E. 
The result by this method may be depended on to the 
third decimal figure, unless the distance is several hundred 
chains, and then it will rarely affect the second decimal 
figure. 

Ex. 2. Required the latitude and departure of a line 
N. 29° 38' E. 26.47 chains. 

29° 38' Cosine .86921 Sine.49445 



Och. 


17.3842 


9.8890 


6 " 


5.2153 


2.9667 


.4 


.3477 


1978 


.07 


608 


346 



Lat. 23.0080 K Dep. 13.0881 E. 

The calculation need not, in general, be carried beyond 
the third decimal place. In the above example the work 
would then stand thus : 

29° 38' 



8' 


Cosine 


.86921 


Sine 


i .49445 


ch. 


17.384 


9.889 


6 " 




5.215 




2.967 


.4 




348 




198 


.07 




61 




34 




Lat. 23.008 K 


Dep. 


13.088 E. 



Ex. 3. Required the latitude and departure of a line bear- 



ing S. 56° 7' E. 63.48 chains. 



Ans. Lat. 35.39 S. ; Dep. 52.70 E. 

Ex. 4. Required the latitude and departure of a line bear- 
ing N". 52° 49' W. 136.75 chains. 

Ans. Lat. 82.65 K ; Dep. 108.95 W. 



Sec. V.] LATITUDES AND DEPARTURES. 199 

Ex. 5. Given the bearing and distance S. 23° 47' W. 
13.62 chains, to find the latitude and departure. 

Ans. Lat. 12.46 S.; Dep. 5.49 "W. 

337. Test of the Accuracy of the Survey. When the 
surveyor has gone round a tract, and has come back to the 
point from which he started, it is self-evident that he has 
travelled as far in a southerly direction as he has in a 
northerly, and as far easterly as westerly. 

His whole northing must equal his whole southing, and. 
his whole easting equal his whole westing. If then the 
north latitudes are placed in one column and the south lati- 
tudes in another, the sum of the numbers in these columns 
will be equal, provided the bearings and distances are 
correct. So also the columns of departures will balance 
each other. 

Owing to the unavoidable errors in taking the measure- 
ments, and also to the fact that the bearings are generally 
taken to quarter degrees, this exact balancing rarely occurs 
in practice. "When the sums are nearly equal, we may 
attribute the error to the want of precision in the instru- 
ments ; but, if the error is considerable, a new survey should 
be made. 

It not unfrequently happens that the mistake has been 
made on a single side. This can often be detected by 
taking the errors of latitude and departure, and calculating 
or estimating the bearing of a line which should produce 
such an error by a mismeasurement of its length or a mis- 
take in its bearing. A little ingenuity will then frequently 
enable the surveyor to judge of the probable position of the 
error, and thus obviate the necessity of a complete resurvey 
of the tract. 

It is laid down as a rule by some good surveyors that an 
error of one link for every five chains in the whole distance 
is the most that is allowable. When the transit or theodo- 
lite is used, a much closer limit should be drawn. One 
link for ten or fifteen chains is quite enough, unless the 
ground is very difficult. Every surveyor will, however, 



200 COMPASS SURVEYING. [Chap. V. 

form a rule for himself, dependent on his experience of the 
precision to which he usually obtains. A young surveyor 
should set a high standard of excellence, as he will find this 
to be a very good method of making himself accurate. If 
he begins by being satisfied with poor results, the chances 
are that he will never attain to a high rank in his profession. 

338. Correction of Latitudes and Departures. 

When the northings and southings, or the eastings and 
westings, do not balance, the error should be distributed 
among the sides before making any calculations dependent 
upon them. 

The usual mode of distributing the error is to apply to 
each line a portion proportioned to its length. 

Rule a table, and head the columns as in the adjoining 
example. Take the latitudes and departures of the several 
sides, and place them in their proper columns. 

Take the difference between the sum of the northings 
and that of the southings. The result is the error in lati- 
tude, and should be marked with the name of the less sum. 

Do the same with the eastings and westings: the result is 
the error in departure, of the same name as the less sum. 

Divide the error of latitude by the sum of the distances : 
the quotient is the correction for 1 chain. 

Multiply the correction for 1 chain by the number of 
chains in the several sides : the products will be the correc- 
tions for those sides, which may be set down in a column 
prepared for the purpose, or at once applied to the 
latitude. 

Operate the same way with the error in departure, to 
obtain the corrections of departure of the several sides. 

The corrections are of the same name as the errors. 

The corrections above found are to be applied by adding 
them when of the same name, but subtracting if of different 
names. 

If one side of a tract is hilly, or otherwise difficult to 
measure, a larger share of the error should be attributed to 
that side. 

When a change of bearing of a long side will lessen the 



Sec. V.] 



LATITUDES AND DEPARTURES. 



201 



error, this change should be made, especially if the survey 
was made with a compass. 

The corrections may be made in the original columns by 
using red ink. Kew columns are, however, to be preferred. 

Ex. 1. Given the bearing and distances as follows, to find 
the corrected latitudes and departures. 



K43|°"W". 

E". 29f ° E. 
S. 80° E. 

East. 

S. 10J° E. 

S. 64° W. 

K 63f ° W. 

S. 57i° W. 



28.43 
30.55 
28.74 
40.00 
23.70 
25.18 
20.82 
31.65 



T 

~2 

IF 
T 

"5 
7T 
T 


Bearings. 


Dist. 


N. 


S. 


E. 


W. 


Cor. 

N. 


Cor. 
W. 

.01 


N. 


S. 


E. 


W. 


N.43^°W. 


28.43 
30.55 


20.62 


— 


"l5l6~ 


19.57 




20.62 




~15JL4~ 
"2T28 


19.58 


N.29%°E. 


26.52 






.02 




S. 80° E. 


28.74 
40.00 




4.99 


28.30 




.01 


.02 
.02 
.01 




4.99 





East. 




23.32 
11.04 


40.00 




.01 




39.98 


S.10i^°E. 


23.70 




4.22 






23.32 


4.21 


22.64~ 


S. 64° W. 


25.18 


_ 9T21" 





22.63 




.01 
.01 

.02 




11.04 





N.63%°W. 


20.82 




18.67 




9.21 




18.68 


8 


S.57^°W. 


31.65 


17.01 




26.69 




17.01 




26.71 




229.07 


56.35 56.36 
56.35 

Er. N. .01 


87.68 87.56 .01 .12 
87.56 

.12 Er. W. 


56.36 


56.36 87.61 87.61 



Ex. 2. Correct the latitudes and departures from the fol- 
lowing notes:— 1. S. 49° W. 12.93 ch.; 2. S. 88° W. 13.68 
ch. ; 3. K 25|° W. 14.09 ch. ; 4. K 43|° E. 14.70 ch. ; 5. 
K 12|° W. 17.95 ch. ; 6. K 88f ° E. 17.68 ch. ; 7. S. 36J° E. 
35.80 ch.; 8. S. 77J° W. 16.15 ch. 

Ans. 1. S. 8.48, W. 9.76; 2. S.. 48, W. 13.67; 3. K 12.73, 
W. 6.01; 4. K 10.70, E. 10.07; 5. K 17.51, W. 3.88; 6. K 
38, E. 17.69; 7. S. 28.79, E. 21.30; 8. S. 3.57, "W. 15.74. 






COMPASS SURVEYING. 



[Chap. V. 



SECTION VI. 



PLATTING THE SURVEY.* 

339. With the Protractor. First Method.— Dn&w a line 
NS, on any convenient part of the paper, to represent the 
meridian. 

Place the protractor with its straight edge to this line, 
and its arc turned to the right if the bearing be easterly, 
but to the left if it be westerly, and with a fine point mark 
off the number of degrees. Draw a straight line from the 
centre to this point, and on it lay off Fi s- 1*3. 

the distance. The point 2 (Fig. 143) 
will thus be determined. Through 2 
draw a line parallel to !N" S. Place the 
protractor with its centre at 2 and its 
straight side coincident with the me- 
ridian, and prick off the degrees in 
the bearing of the second side. Join 
this point to 2, and on the line thus 
determined lay off 2.3 equal to the 
second side. Through 3 draw another 
meridian ; and so proceed until all the 
bearings and distances have been laid down. 

When the last line has been platted, it should end at the 
starting point: if it does not, either the notes are incorrect 
or an error has been made in the platting 

The proper position of the protractor after the first may 
be determined without drawing meridians, by placing the 
centre at the point and turning the protractor until the 
number of degrees in the bearing of the last line coin- 
cides with that line. Its position is then parallel to the 
former one, and the bearing of the next line may be 
pricked off. 

This method is the one commonly employed. It has, 
however, the disadvantage of accumulating errors, since any 
mistake in laying down the bearing of one line will alter 




* Various hints in this section have been derived from Gillespie's " Land 
Surveying." 



Sec. VI.] 



PLATTING THE SURVEY, 



203 



both the direction and position of every subsequent line on 
the plat. 

The figure is the plat from the following field-notes :— 
2. S. 60J° E. 10.80; 3. S. 8° E. 9.50; 



1. K 27|° E. 7.75 
4. S. 47J° E. 9.37 



5. S. 54J° W. 8.42 



6. K 37J° "V7. 23.69. 




340. Second Method. — Draw a number of parallel lines to 
represent meridians. They may be equidistant or not. 
The faint lines on ruled paper will answer very well. 

Select any convenient point for Kg. 144. 

a place of beginning, and draw the 
line AB (Fig. 144) for the first side. 
Place the protractor so that its 
centre shall be on one of the me- 
ridians, and turn it until the num- 
ber of degrees in the next side 
coincides with the same meridian, 
as at C : slip it down the line, 
maintaining the coincidence of the 
centre and degree mark with the meridian, until the 
straight side passes through the point Draw a line along 
this side. It will be the direction of the required line, on 
which lay off the given distance. So continue until all the 
sides have been platted. The figure will close, if the work 
is properly done. 

This method is quite as accurate as the last, and admits 
of very rapid execution. 



341. By a Scale of Chords. With 
a radius equal to the chord of 60° 
describe a circle near the middle of 
the paper. Through its centre O (Fig. 
145) draw a line NS to represent the 
meridian. Lay off from the north and 
south points the different bearings, 
marking them 1, 2, &c. Through 
A, any convenient point, draw AB 
parallel to 0.1, and on it lay off AB 
equal to the length of the first side 




204 



COMPASS SURVEYING. 



[Chap. Y. 



taken from any convenient scale. Through. B draw BC 
parallel to 0.2: on it lay off BC equal to the second side. 
Through C draw CD parallel to 0.3; and so proceed till all 
the lines have been platted. 

With an accurate scale of chords of a good size, this 
method is probably preferable to either of the others. The 
scale on the rule sold with cases of instruments, however, 
is so small that no- great precision can be obtained by its 
use. It is still, however, preferable to the other methods if 
the protractor in similar cases of instruments is employed. 



Fig. 146. 

N 



342. By a Table of Natural Sines. The sine of any 
arc is equal to half the chord of twice that arc, or to the 
chord of twice the number of degrees on a circle of half 
the radius. We may therefore use a table of natural sines 
to lay off angles. Its use in protracting a survey is ex- 
plained below. 

Describe a circle (Fig. 146) 
about the centre of the paper 
with a radius equal to 5 on a 
scale of equal parts. This scale 
should be taken as large as con- w - 
venient. Through its centre A 
draw NS to represent the me- 
ridian, and cross the circle at the 
points marked 60°, with the 
centres N* and S, and radius equal to that of the circle: also 
draw EW perpendicular to NS. The points marked 30° 
may be obtained by crossing the circle with the compasses 
opened to the radius and one leg at E and W. 

A skeleton protractor is thus formed, having the North, 
South, East, and West points, as well as the 30° and 60° 
points, accurately laid down. 

Commencing with the first bearing, which in the figure is 
!N\ 27| E., divide it by 2, and from the table of natural 
sines take out the sine of the quotient 13° 45'. It is fouud 
to be 2.3769, the decimal point being removed 1 place to 
the right. Take this distance 2.38 from the scale of equal 
parts, and lay it off from 1ST to 1. 




Sec. VI.] PLATTING THE SURVEY. 205 

The second bearing is S. 60J° E. The half of J° is 15': 
the sine of this is 0.0436. Lay off .04 from 60° to 2. 

The third bearing is S. 8° E. : the sine of 4° is 0.6976. Lay 
off .70 from S. towards E. : the point 3 is thus determined. 

The fourth is S. 47^° E., which exceeds 30° by 17|° : the 
half of 17J° is 8° 45', of which the sine is 1.5212. 1.52 
laid off from 30 towards E. determines the point 4. 

An accurate protractor is thus formed on the paper, con- 
taining all the bearings in the field-notes. The subsequent 
work will be as in last article. 

343. By a Table of Chords. Instead of a table of 
natural sines, a table of chords, when it can be procured, is 
more convenient. 

Prepare a circle, as in last article, with the !N\, S., E., W., 
and the 30° and 60° points, the radius being 10, taken 
from a scale of equal parts. 

Take from the table the chord of the number of degrees, 
or of its excess above 30° or 60°, and lay it off from the 
proper point, as directed in last article: an accurate pro- 
tractor is thus formed on the paper, and the work proceeds 
as before. 

The object in determining the 30° and 60° points is to 
avoid the necessity of laying off long distances. When the 
compasses are much stretched, the points strike the paper 
very obliquely, and are apt to sink in so as to make the dis- 
tance laid off slightly too short. 

This method is preferable to any of those which precede 
it : it is only to be excelled by the one next given. 

344. By Latitudes and Departures. 

Where the latitudes have been calculated and balanced, 
they afford the most convenient and accurate means of 
platting the survey. 

Rule five columns, heading them Sta., jN"., S., E., W, 
Commencing at any convenient station, place the latitude 
and departure of the side beginning at this station oppo- 
site the next station in the table, and in their appropriate 
columns. When the latitude set down is of the same name 



206 



COMPASS SURVEYING. 



[Chap. V. 



as that of the next side, add them together, and place the 
result in the proper column of latitudes opposite the next 
side. But if they be of different names, take their differ- 
ence, and place it in the column of the same name as the 
greater. Proceed in the same way with this result and the 
next latitude, and so continue till all the latitudes have 
been used. The results will be the latitude of the stations 
opposite which they are placed, all counted from the point 
at which we commenced. 

Proceed in the same manner with the departures. Thus, 
if it were required to plat the survey of which the field- 
notes are given Ex. 1, Art. 338, we have the latitudes and 
departures, as in the following table. (See the example re- 
ferred to): — 



Sta. 


N. 


S. 


E. 


w. 


1 


20.62 






19.58 


2 


26.52 




15.14 




3 




4.99 


28.28 




4 


.01 




39.98 




5 




23.32 


4.21 




6 




11.04 




22.64 


7 


9.21 






18.68 


8 


• 


17.01 




26.71 



Preparing a table as above directed, and beginning at the 
fourth station, the total latitudes and departures will be as 
below : — 



Sta. 


N. 


s. 


E. 


w. 


1 




42.15 




23.84 


2 




21.53 




43.42 


3 


4.99 






28.28 


4 


00 






0.00 


5 


.01 




39.98 




6 




23.31 


44.19 




7 




34.35 


21.55 




8 




25.14 


2.87 





Sec. VI.] 



PLATTING THE SURVEY. 



207 



The latitude of the fourth side is .01 N". This is put in 
the column headed north, opposite the fifth station. The 
next latitude being south, take the difference 23.31 ; place 
it in the south : add 23.31 and 11.04, both being south, and 
we have 34.35 S. Subtract from this 9.21 K leaves 25.14 S. 
This, added to 1T.01 S., gives 42.15 S. , Subtract 20.62 K 
leaves 21.53 S.; 21.53 S. from 26.52 K, the next latitude, 
leaves 4.99 K Finally, 4.99 N. and 4.99 S. cancel, leaving 
for the latitude of the fourth station. In the same man- 
ner we find the total departures. 

As the latitude and departure of the station with which 
we begin are zero, the work proves itself. It is usual to 
begin with the first side. 

The table having been prepared as above, draw on any 
convenient part of the paper a meridian line, NS, (Fig. 147,) 
and take any point E for the starting point. From this 




point, lay off the several total latitudes contained in the table 
above or below the point as the latitude is north or south, and 
number them according to the station to which they are op- 
posite in the table. 

Through these points draw perpendiculars to the me- 
ridian, and make them equal to the several total de- 
partures, — laying the distance to the right hand if the 
departure be east, but to the left if it be west. The cor- 



208 



COMPASS SURVEYING. 



[Chap. V. 



ners will thus be determined. When these are joined, the 
plat will be completed. 



SECTION VII. 

PROBLEMS IN COMPASS SURVEYING. 

345. Problem 1. — Given the bearing of one side, and 
the deflection of the next, to determine its bearing. 

If the given bearing is northeasterly or southwesterly, add 
the deflection if it is to the right hand. If the sum exceeds 
90°, take its supplement, and change north to south, or 
south to north. 

If the deflection is to the left hand, subtract it from the 
bearing ; but if it is greater than the bearing from which it 
is to be subtracted, take the difference, and change east to 
west, or west to east. 

"When the given bearing is northwesterly or southeasterly, 
add the left-hand and subtract the right-hand defections, ap- 
plying the same rules as above. 

Examples. 

Ex. 1. Given AB (Fig. 148) K 37° E., 
and the deflection of the next side 43° 
1 5 r to the right. 

BD=K 37° E. w . 
DBC = 43° 15' 
Whence BC is K 80° 15' E. 

Ex. 2. Given AB K 37° E., and the deflection of BC 
43° 15' to the left. 

BD = N. 37° E. 
DBC = 43° 15 ' 
Whence BC' is K 6° 15' W. 




Sec. VII. ] PROBLEMS IN COMPASS SURVEYING. 209 

Ex. 3. Given the bearing of AB, K 39° W., and BC de- 
flects to the left 75° 26': required the bearing of BC. 

Ans. S. 65° 34' W. 

Ex. 4. Given the bearing of a line S. 63° 29' E., and 

the deflection of the next 29° 17' to the right : required its 

bearing. 

Ans. S. 34° 12' E. 

Ex. 5. The bearing of one line being S. 34° 12' E., and 
the deflection of the next 75° 32' to the right: required its 
bearing. 

Ans. S. 41° 20' "W. 

346. Problem 2. — To determine the angle of deflection 
between two courses. 

1. If the lines run between the same points of the com- 
pass, take the difference of their bearings. 

2. If they run between points directly opposite, subtract 
the difference of the bearings from 180°. 

3. If they run from the same point towards different 
points, add the bearings. 

4. If they run from different points towards the same 
point, take the sum of the bearings from 180°. 

Examples. 

Ex. 1. AB (Fig. 149) runs S. 56° W., ^149. 

and BC S. 25° "W\ : required the de- 
flection. 



w 



56° 

25° 

Deflection 31° to the left. 

14 




210 



COMPASS SURVEYING. 



| Chap. V. 



Ex. 2. Given AB (Fig. 150) K 46 W., 
and BC S. 79° E.: required the de- D 
flection. 

K 46° W. 

S. 79° E. 



Fig. 150. 

N 



AB 
BC 

ABC 



DBC 



33° 
180° 
147° = deflection to the right. 




Ex. 3. Given AB (Fig. 151) K 39° E., 
and BC K 63° W., to find the de- c 
flection. 

AB K 39° E. 

BC K 63° W. 

DBC 102° = deflection to the left. 



Fig. 151. 

N 




Ex. 4. Given AB (Fig. 152) S. 82° E., 
and BC K 67° E., to find the de- 
flection. 

AB S. 82° E. 

BC K 67° E. 



Fig. 152. 

N 



DBC 



149° 
180° 
31° = deflection to the left. 




Ex. 5. The bearing of a line is K 46° 30' E., and that of 
the next S. 63° 29' "W. : required the deflection. 

Ans. 163° V to the left. 

Ex. 6. "What is the deflection in passing from a course 

S. 63° W. to one K 29° W. ? 

Ans. 88° to the right. 

Ex. 7. "What is the deflection in passing from a course 

K 82i W.to one K 29J° W.? 

Ans. 53|° to the right. 

347. Angle between lines. If the angle between two 



Sec. VII.] PROBLEMS IN COMPASS SURVEYING. 211 

lines is required, reverse the first bearing, and apply the 
above rules. 

Examples. 

Ex. 1. Given AB K 87° E., and BC S. 25° W., to find 
the angle ABC. Ans. ABC = 62°. 

Ex. 2. Given AB S. 63° E., and BC K 56° E.: required 
the angle ABC. Ans. ABC = 119°. 

Ex. 3. Given CD K 15° \Y. 5 and DE X. 56° W.: required 
the angle CDE. Ans. CDE = 139°. 

Problem 3. — To change the bearings of the sides of a 
survey. 

348. It is frequently useful to change the bearings of a 
survey so as to determine what they would be if one side 
were made a meridian. This change is made on the sup- 
position that the whole plat is turned around without alter- 
ing the relative positions of the sides. Every bearing will 
thus be altered by the same angle. The following rules 
take in all the possible cases. 

The reason of these rules will be made apparent by 
drawing a figure to represent any particular case. 

1. Deduct the bearing of the side that is to be made a 
meridian from all those bearings that are between the same 
points as it is, and also from those that are between points 
directly opposite to them. If it is greater than any of 
those bearings, take the difference, and change west to east, 
or east to west. 

2. Add the bearing of the side that is to be made a 
meridian to those bearings that are neither between the 
same points as it is, nor between points directly opposite. 
If either of the sums exceeds 90°, take the supplement, and 
change south to north, or north to south. 

Examples. 
Ex. 1. The bearings of a tract of land are, — 1. N". 57° E.; 



212 



COMPASS SURVEYING. 



[Chap. V. 



2. K 89° E.; 3. S. 49|° E.; 4. South; 5. S. 27|° W.; 6. 
S. 53i° W. ; 7. JNT. 89° W. ; 8. K 37° W. ; 9. K 43° E. to 
the place of beginning. Kequired to change the bearings, 
so that the ninth side may be a meridian. 



1. K 57° E. 
K 43° E. 



2. K 89° E. 
K 43° E. 



3. S. 49J° E. 
K43° E. 





If. 14° E. 




IF. 46° E. 


92J° 
180° 
K 87J° E. 


4. 


s. o°w. 


5. 


S. 27f ° W. 


6. S. 53J° W 




K 43° E. 




K43° E. 


K 43 ° E. 




S. 43° E. 




S. 15J° E. 


S. 10i° w 


7. 


K 89° W. 
K 43° E. 
132° 

180° 


8. 


K". 37° W. 
K 43° E. 
K 80° W. 


9. North. 



S. 48° W. 



Ex. 2. Change the bearings in the following notes, so 
that the second side may be a meridian : — 1. N. 43° 25' W. ; 
2. 1ST. 29° 48' E. ; 3. 8. 80° E. ; 4. K 89° 55' E. ; 5. S. 10° 
13' E.; 6. K 63°55'W.; 7. S.63°45'W.; 8. N. 57°35'W. 

Ans. 1. K 73° 13' W.; 2. North; 3. N. 70° 12' E.; 

4. K 60° 7' E. ; 5. S. 40° V E.; 6. S. 86° 17' W.; 7. 

5. 33° 57' "W. ; 8. N. 87° 23' W. 

Ex. 3. Change the bearings in the following notes, so 
that the fourth side may be a meridian : — 1. S. 63° E. ; 2. 

5. 47° E. ; 3. S. 59i° W. ; 4. N. 84i° W. ; 5. 25T. 12° W. ; 

6. N. 17i° E., and 7. S. 29f ° W. 

Ans. S. 21i° W.; % S. 37i°"W.; 3. N. 36| W.; 4. 
North; 5. K 72J° E.; 6. S. 78° E.; 7. N. 65 j° W. 



Sec. VIIL] SUPPLYING OMISSIONS. 213 



SECTION VIIL 

SUPPPLYING OMISSIONS. 

349. When any two of the dimensions have been omit- 
ted to be taken, or have become obliterated from the field- 
notes, these may be supplied. This should never lead the 
surveyor to neglect to take every bearing and every dis- 
tance. It is far better to use almost any means, however 
indirect, to obtain all the bearings and distances indepen- 
dently of one another than to determine any one. from the 
rest. If one side is determined from the others, all the 
errors committed in the measurements are accumulated on 
that side, and thus the means of proving the work by the 
balancing of the latitudes and departures is lost. The 
various problems in Section 3 will enable the young sur- 
veyor to solve almost every case of difficulty that will be 
likely to occur in making his measurements. Should any 
difficulty arise to which none of the methods there de- 
veloped are applicable, a knowledge of the principles of 
Trigonometry will afford him the means of overcoming it. 

CASE 1. 

350. The bearings and distances of all the sides except 
one, being given, to determine these. 

Determine the latitudes and departures of those sides of 
which the bearings and distances are given. Take the 
difference between the sums of the northings and southings, 
and also between the sums of the eastings and westings : 
the remainders will be the latitude and departure of the 
side the bearing and distance of which are unknown. 
With this latitude and departure calculate the bearing and 
distance by Art. 333. 

This principle will enable us to determine a side when it 
cannot be directly measured. Thus, run a series of courses 
and distances, so as to join the two points to be connected. 



214 



COMPASS SURVEYING. 



[Chap. V. 



These, with the unknown side, form a closed tract, the 
sides of which are all known except one. 

It will likewise enable us to determine the course and 
distance of a straight road between two points already 
connected by a crooked one. In both these cases it is best, 
where the nature of the ground will admit of it, to run the 
courses at right angles to each other, as in 
Fig. 153, in which AB is the distance to 
be determined. Run AC any direction, 
CD perpendicular to AB, DE to CD, EF 
to DE, FG to EF, and, finally, GB per- 
pendicular to F.G through B. 

Then, assuming AC as a meridian, AC 
+ DE + FG will be the latitude of AB 
and CD + EF + GB the departure. From 
these calculate the distance AB and the 
bearing BAC. This angle applied to the 
true bearing of AC will give that of AB. 




Examples. 

Ex. 1. The bearings and distances of the sides of a tract 
of land being as follows, it is desired to find the bearing 
and distance of the third side, — viz. : 1. N. 56 J° "W. 15.35 
chains; 2. K 9° W. 19.51 ch.; 3. Unknown; 4. S. 39}° E. 
13.35 ch.; 5. K 82 J° E. 12.65 ch.; 6. S. 6|° W. 12.18 ch.; 
7. S. 52i° W. 20.95 ch. 






Sec. VIII.] 



SUPPLYING OMISSIONS. 



215 



Sta. 


Bearing. 


Distance. 


N. 


s. 


E. 


w. 


1 

~2~ 


K 56i° "W. 


15.35 


8.53 






12.76 


K 9° "W. 


19.51 


19.27 






3.05 


3 












4 


S. 39|° E. 


13.35 




10.26 


8.54 




5 


K 82J° E. 


12.65 


1.65 


1L2JL0" 


12.54 




6 


S. 6£° W. 


12.18 





1.43 


7 


S. 52J° W. 


20.95 


12.75 




16.62 








29.45 


35.11 


21.08 


33.86 



29.45 

5.66 E". 



21.08 
12.78E. 



Diff. Lat. 

Departure, 

Bearing, 

Bearing, 
Diff. Lat. 
Distance, 



5.66 


log. 


0.752816 


12.78 


log. 


1.106531 


HT. 66° V E. 


tang. 


10.353715 


66° V 


cos. 


9.607322 




log. 


0.752816 



13.98 



1.145494 



Ex. 2. One side AB of a tract of land running through a 
swamp, it was impossible to take the bearing and distance 
directly. I therefore took the following bearings and dis- 
tances on the fast land,-viz. : AC, K 47° W. 16.55 chains; 
CD, K 19° 5' E. 11.48 ch. ; DE, 1ST. 11° 5' W. 15.53 ch.; 
EF, K 23° E. 9.72 ch., and EB, K. 75° 12 r E. 14.00 chains. 
Eequired the bearing and distance of AB. 



216 



COMPASS SURVEYING. 



[Chap. V. 



Sta. 


Bearing. 


Distance. 


N. 


s. 


E. 


w. 


A 


K 47° W. 


16.55 


11.29 






12.10 


C 


1ST. 19° 5' E. 


11.48 
~15J53 


10.85 
~1572T 
_ 8795" 




3.75 




D 


K11°5'W. 




2.99 


K 23° E. 


9.72 




3.80 
~137oT 




K75°12'E. 


14.00 


3.58 


(49.91) 




B 






~2L09~ 


(6.00) 








49.91 




15.09 



15.09 







6.00 


Diff. Lat. 


49.91 


log. 1.698188 


Departure, 


6.00 


log. 0.778151 


Bearing AB, 


K 6° 51' E. 


tang. 9.077963 


Bearing, 


6° 51' 


cos. 9.996889 


Diff. Lat. 


50.27 


1.698188 


Distance, 


1.701299 



Note. — In calculations of this kind, it is sufficiently accurate to confine the 
operations to two decimal places, unless the number of sides is large. In Ex. 
2, had the work been extended to the third decimal place, it would not have 
made more than 15 // difference in the bearing and 1 link in the distance. 

Ex. 3. Given the bearings and distances as follows, — viz. : 
1. S. 29f ° E. 3.19 ; 2. S. 37i° W. 5.86 ; 3. S. 39J° E. 11.29 ; 

4. H". 53° E. 19.32; 5. Unknown; 6. S. 60}° W. 7.12; 7. S. 
29J°E. 2.18; 8.S. 60J o 'W.8.12; to find the bearing and dis- 
tance of the fifth side. Ans. K". 31° 5' W. 16.26 ch. 

Ex. 4. Required the bearing and distance of the third 
side from the following notes:— 1. K 46° 40' W. 18.41 
chains; 2. K 54J E. 13.45 chains; 3. Unknown; 4. S. 
74° W E. 17-58 chains; 5. S. 47° 50' E. 15.86 chains ; 6. 

5. 47° 25' W. 16.36 chains ; 7. S. 62° 35' W. 14.69 chains. 

Ans. 3d side, K 5° 26' W. 12.67 ch. 

Ex. 5. It being impossible to take the bearing and dis- 
tance of one side AB of a tract of land directly, in con- 



Sec. VIII.] SUPPLYING OMISSIONS. 217 

sequence of a marsh grown up with thick bushes, I took 
bearings and distances on the fast land as below, — viz. : 
AC S. 49i° W. 9.30 chains ; CD S. 32|° E. 10.25 chains ; DE 
S. 5J° W. 6.75 chains ; and EB K 79|° E. 8.10 chains. Ke- 
quired the bearing and distance of the side AB. 

Ans. S. 16° 12' E. 20.82 ch. 

Ex. 6. The bearings and distances taken along the middle 
of a road which it is desired to straighten are as below, — 
1. S. 27° 30' E. 12.65 chains; 2. S. 10|° E. 23.45 chains; 3. 
S. 14° W. 124.33 chains; 4. S. 67° E. 82.43 chains; 5. S. 17° 
E. 96.35 chains. Required the bearing and distance of a 
new road that shall connect the extremities. 

Ans. S. 16°44'E. 291.63 ch. 

CASE 2. 

351. The bearings and distances of the sides of a tract 
of land being given, except two, — one of which has the 
bearing given, and the other the distance and the points 
between which it runs, — to determine the unknown beariDg 
and distance. 

Rule. 

Change the bearings so that the side whose bearing only 
is given, may be a meridian. 

Take out the latitudes and departures according to these 
changed bearings. Take the difference of the eastings and 
westings: this difference will be the departure of the side 
not made a meridian. 

"With this departure and the given distance, calculate by 
Art. 333 the changed bearing and difference of latitude, 
and place the latter in the column of latitude. From the 
changed bearing the true bearing may readily be found. 

Take the difference between the northings and south- 
ings. This difference is the difference of latitude of the 
side made a meridian, and is equal to the distance. 

Note. — In general, there will be no difficulty in determining whether the 
changed bearing found should be north or south. In some cases, however, 
either will render the true bearing conformable to the points given. In this 
case the question is ambiguous, and can only be determined from the other 
data, except when the true bearing is nearly known. 



218 



COMPASS SURVEYING. 



[Chap. V. 



Examples. 

Ex. 1. Given the courses and distances as below, to find 
the unknown bearing and distance. 



Sta. 
1 

2 
3 
4 

5 
6 

7 


Bearing. 


Changed 
Bearing. 


Dist. 


N. 


s. 


E. 


w. 


K 56% W. 


S. 57f W. 


15.35 




8.19 




12.98 


N. 9 W. 


1ST. 75 "V7. 
"T^orth. 


19.51 


5.05 




18.85 


1ST. 66 E. 




(14.00) 








S. 39 j E. 


N. 74J E. 


13.35 


3.62 




12.85 




1ST. E. 
~S. 6f W. 

"sr"52JW7 




12.65 


(12.12) 




(3.62) 




S. 59£ E. 


12.18 

20.95 




6.23 


10.47 




S. 13J e. 




20.37 


4.89 


31.83 










34.79 


34.79 


31.83 



Dist., fifth side, 12.65 

Dep. " 3.62 

Ch. bear. " 2SL 16° 38' E. 

66° 



A. C. 8.897909 
0.558709 



sin. 9.456618 



N". 82° 38' E., bearing of fifth side. 

Ch. bear., fifth side, 16° 38' 
Dist. " 

Diff. Lat. " 12.12 

Dist., third side, 14.00 ch. 



cos. 9.981436 
1.102091 

"1.083527 



Ex. 2. Given— 1. K 47° W. 16.55 chains; 2. K 19° 5' W. 

11.48 chains; 3. I". W. 15.53 chains ; 4. K 23° E. 9.72 

chains; 5. K 75|° E. 14 chains; 6. S. 7° E., unknown; 
to determine the bearing of the third and the distance of 
the sixth side. 

Ans. 3d side, K 28J° W. ; 6th, 48.67 ch. 



Sec. VIII. ] 



SUPPLYING OMISSIONS. 



219 



CASE 3. 

352. The bearings and distances of the sides of a tract 
of land being given, except the distances of two sides, to 
determine these. 

Rule. 

Change the bearings so that one of the sides the dis- 
tance of which is unknown may be a meridian. Take out 
the latitudes and departures with these changed bearings. 
The difference of the eastings and westings will be the de- 
parture of the side not made a meridian. With this de- 
parture and the changed bearing, find the distance and 
difference of latitude. Place the latter in its proper place 
in the table. Take the difference between the northings 
and southings: this difference will be the difference of 
latitude of the side made a meridian, and will be equal to 
the distance. 

Examples. 

Given as follow,— 1. K 56%° W. 15.35 chains ; 2. K 9° W., 
unknown; 3. K 66° E. 14.00 chains; 4. S. 39 j° E. 13.35 
chains; 5. K 82}° E., unknown ; 6. S. 6f¥, 12.18 chains; 
7. S. 52J° W. 20.95 chains; to find the distances of the 
second and fifth sides. 



Sta. 
1 


Bearing. 


Changed 
Bearing. 


Dist. 


N. 


s. 


E. 


w. 


K56JW. 


K 47JW. 


15.35 


10.42 






11.27 


2 
3 


K 9 W. 


North. 


(19.54) 


(19.54) 








1ST. 6Q E. 


K 75 E. 


14.00 


3.62 




13.52 




4 

5 


S. 39f E. 


S. 30f E. 


13.35 




11.47 


6.83 





K82fE. 


S. 88| E. 






.39 


(12.64) 


6 

7 


S. 6f W. 


S. 15} W. 


12.18 




11.72 




3.31 

18.41 
32.99 


S.52JW. 


S. 61J W. 


20.95 




10.00 












33.58 


33.58 


32.99 



220 COMPASS SURVEYING. [Chap. V. 

Ch. bear., fifth side, 88° 15' A. C. sin. 0.000203 
Dep. " 12.64 1.101747 

Dist. " 12.65 1.101950 

Ch. bear. cos*. 8.484848 

Dist. 1.101950 



Diff. Lat. 0.39 S. — 1.596798 

Ex. 2. Given— 1. S. 29f° E. 3.19 chains; 2. S. 37J° W. 
5.86 chains; 3. S. 39J° E., unknown; 4. K 53° E. 19.32 
chains; 5. K 31° 5' W., unknown; 6. S. 60f° W. 7.12 
chains; 7. S.29|°E.2.18 chains; 8. S. 60J° W. 8.12 chains ; 
to find the distances of the third and fifth sides. 

Ans. 3d side, 11.28 chains; 5th, 16.26 chains. 

CASE 4. 

353. The bearings and distances of all the sides of a tract 
of land being known except the bearings of two sides, to 
determine these. 

Rule. 

Take out the differences of latitude and the departures 
of the sides whose bearings and distances are known. The 
differences of the northings and southings will be the dif- 
ference of latitude, and that of the eastings and westings 
the departure, of a line which, with the known sides of the 
survey, will form a closed figure, and may therefore be called 
the closing line. 

With this closing line and the distances of the two other 
sides form a triangle. 

Calculate two angles of this triangle. These angles 
applied to the bearing of the closing line will give the 
bearings required. 



Sec. VIII.] 



SUPPLYING OMISSIONS. 



221 



Examples. 

Ex. 1. Given AB (Fig. 154) K 56J° W. 15.35 chains; BC 
K 9° W. 19.51 chains; CD K — E. 14 chains ; DE S. 39|° 
E. 13.35; EF 1ST. 82|° E. 12.65 chains; FG S. — W. 12.18 
chains; GA S. 52J° W. 20.95 chains ; to find the bearings 
of the third and sixth sides. 



AB 
BC 

Ce 

ef 
GA 


Bearing. 


Dist. 


N. 


s. 


E. 


w. 


N". 56i w. 


15.35 


8.53 







12.76 


K". 9¥. 


19.51 


19.2T 




3.05 


S. 39| E. 


13.35 
12.65 


1.65 


10.26 


8.54 




K 82J e. 




12.54 


' 


S. 52i w. 


20.95 


12.75 




16.62 








29.45 


23.01 


21.08 


32.43 



23.01 



21.08 







6.44 


11.35 


Diff. Lat. 




6.44 


A. C. 9.191114 


Dep. 


, 


11.35 


1.054996 


Tang, closing 


line, 


S. 60° 26' E. 


10.246110 


Cos. bear. 




60° 26' 


A.C. 0.306769 


Diff. Lat. 






0.808886 


Dist. closing 


line, 


13.05 


1.115655 


FG 




12.18 




/G 




13.05 


A. C. 8.884388 


/F 




14.00 
2)39.23 


" " 8.853872 






19.615 


1.292588 






7.435 


0.871281 




2)19.902129 


JF/G 




26° 41' 


cos. 9.951064 


F/G 




53° 22' 





222 COMPASS SURVEYING. [Chap. V. 



FG 




12.18 


A. C. 8.914353 


/F 




14.00 


1.146128 


sin. F/G 




53° 22' 


9.904429 


sin./GF 




67° 17' 


9.964910 






60° 26' Bear 


of/G 




S 


6° 51' W. 


" GF 



180° - (53° 22' + 60° 26') = 66° 12'; 
therefore, K 66° 12' E. is the bearing of CD. 

Ex. 2. Given— 1. S. 29f° E. 3.19 chains; 2. S. 37J° W. 
5.86 chains; 3. S. — E. 11.29 chains; 4. K 53° E. 19.32 
chains; 5. K — W. 16.26 chains; 6. S.60f°W. 7.12 chains; 
7. S. 29J° E. 2.18 chains ; 8. S. 60J° W. 8.12 chains ; to 
find the bearing of the third and fifth sides. 

Ans. 3d side, S. 39° 8'E.; 5th, K 31° W. 

354. The first three of the preceding rules are so simple 
as hardly to need any explanation. The principle of the 
last will be seen from the following illustration. The figure 
being protracted from the field-notes in Ex 1, Case 4, these 
are, as will be seen, the same as Ex. 1 in the other 
cases. _. t _. 

Fig. 154: 

Let ABCDEFG (Fig. 154) be the d 

plat of the tract, the bearings of CD ,^^^\ 
and FG being supposed unknown. f\ 
If Ce and ef be drawn parallel to 1 \^^**_ 
the sides DE and EF, and /G be 1 ^"" 
joined, then will ABCe/G form a B V 
closed figure, the bearings and dis- ^^ J yS 
tances of all the sides except /G *a 

being known. The course and dis- * 

tance of this side, which is the closing line, are found as 
directed in the rale. Join /F and eE. Then /F is equal 
and parallel to eE and therefore to CD. The sides of the 
triangle /FG are therefore the closing line, the side FG, 
and the line /F equal and parallel to the side CD. In/FG 
find the angles /and G: these applied to the bearing of/G 
will give the bearings of /F or CD and of FG. 



Sec. IX.] CONTENT OP LAND. 223 

This method might have been employed in Cases 2 and 3. 
Those given in the rules are, however, more concise, and 
are therefore to be preferred. 

355. Though the methods illustrated above will serve to 
supply omissions in all cases where not more than two of 
the dimensions are unknown, yet it will not be amiss again 
to impress on the young practitioner the necessity, in all 
cases in which it is practicable, of determining each side 
independently of every other. The rules for supplying 
omissions should only be used in cases where one or more 
of the data have been accidentally omitted, or have become 
defaced on the notes. However accurate the field-work may 
be, there is always a liability to error, and if one side is 
determined by the rest no means are left of detecting any 
error. When a side cannot be measured directly, the best 
way is to determine it by some of the trigonometrical 
methods, taking the angles and base-lines with great care. 
In this way a degree of accuracy may be obtained equal to 
that of the sides measured directly. The latitudes and de- 
partures may then be balanced as usual. 



SECTION IX. 
CONTENT OF LAND. 



356. From the bearings and distances of the sides of a 
tract of land, or from the angles and the lengths of the 
sides, the area may be found, however numerous the sides 
may be. This may be done by Problem 4, which is entirely 
general, it being applicable whatever the number of sides 
may be, provided they are straight lines. As, however, 
there are other more concise methods applicable to triangles 
and quadrilaterals, those are first given. 

If one or more of the boundaries is irregular, instead of 
multiplying the number of sides by taking the bearings of 



224 



COMPASS SURVEYING. 



[Chap. V. 



all the sinuosities of the boundary, it is better to run one or 
more base lines and take offsets', as directed in chain sur- 
veying. The content within the base lines is then to be 
calculated, and the area cut off by the base lines, being 
found by the method Art. 256, is to be added to or sub- 
tracted from the former area, according as the boundary is 
without or within the base. 

As has been already remarked, (Art. 257,) when the tract 
bounds on a brook or rivulet, the middle of the stream is 
the boundary, unless otherwise declared in the deed. Lands 
bordering on tide water go to low-water mark. When the 
stream, though not tide water, is large, the area is generally 
limited by the low-water mark, or by the regular banks of 
the stream. 

If the farm bounds on a public road, the boundary is, 
except in special cases, the middle of the road, and the 
measures are to be taken accordingly. 



357. Problem 1. — Given two sides and the included angle 
of a triangle or parallelogram, to determine the area. 

Say, As radius is to the sine of the included angle, so is 
the rectangle of the given sides to double the area of the tri- 
angle, or to the area of the parallelogram. 

Demonstration. — We have, (Fig. 155,) by 
Art. 137, — 

As rad. : sin. A : : AC : CD : : AB . AC : AB . 
CD, (Cor. 1.6) ; but AB . CD = 2 ABC. 

Examples. 

Ex. 1. Given AB = 12.36 chains, 
BC = 14.36 chains, and ABC = 47° 35', to determine the 
area of the triangle. 




As rad. 

sin. B 

AB 

BC 

2 ABC 



{ 



47° 35' 
12.36 ch. 
14.36 
2)131.033 



AC. 0.000000 
9.868209 
1.092018 
1.157154 
2.117381 



65.5165 ch. = 6 A., 2 R., 8.26 P. 



Sec. IX.] CONTENT OF LAND. 225 

Ex. 2. Given AB K 37° 14' W. 17.25 chains, and BC 
K 74° 29' W. 10.87 chains, to determine the area of the 
triangle ABC. Ans. 5 A., 2 K,, 28 P. 

Ex. 3. Given AB = 23.56 chains, AC = 16.42 chains, and 
the angle A 126° 47'. Required the area of the triangle. 

Ans. 15 A., 1 R., 38.7 P. 

358. Problem 2. — The angles and one side of a triangle 
being given, to determine the area. 

Say, As the rectangle of radius and sine of the angle op- 
posite the given side is to the rectangle of the sines of the 
other angles, so is the square of the given side to double 
the area. 

Demonstration. — We have (Fig. 155) 

t : sin. A : : AC : CD (Art. 137), 
and sin. B : sin. C : : AC : AB (Art. 139). 

(23.6) r . sin. B : sin. A. sin. C : : AC 2 : AB . CD, or 2 ABC. 

Examples. 
Ex. 1. Given AB = 21.62 chains, and the angle A= 47° 
56' and B = 76° 15', to find the area. 

As 



Trad, 
jsin. C 




A.C 


. 0.000000 


55° 49' 


tc 


0.082366 


rsin. A 
(sin. B 


47° 56' 




9.870618 


76° 15' 




9.987372 


JAB 

Iab 


21.62 ch. 




1.334856 


21.62 




1.334856 


2 ABC 


2 ) 407.444 




2.610068 



Area = 203.722 ch. = 20 A., 1 E., 19.5 P. 



c 



Ex. 2. Given AB 17.63 chains, and the angle A = 63 
52' and B 73° 47', to find the area. 

Ans. 19 A., 3 R, 22 P. 

Ex. 3. Given one side 15.65 chains, and the adjacent 
angles 63° 17' and 59° 12', to determine the area of the 
triangle. Ans. 11 A., R, 22 P. 

15 



226 



COMPASS SURVEYING. 



[Chap. V. 



359. Problem 3. — To determine the area of a trapezium, 
three sides and the two included angles being given. 



Rule. 

1. Consider two adjacent sides and their contained angle 
as the sides and included angle of a triangle, and find its 
double area by Prob. 1. 

2. In like manner, find the double area of a triangle of 
which the two other adjacent sides and their contained 
angle are two sides and the included angle. 

3. Take the difference between the sum of the given 
angles and 180°, and consider the two opposite given sides 
and this difference as two sides and the included angle of a 
triangle, and find its double area. 

4. If the sum of the given angles is greater than 180°, 
add this third area to the sum of the others ; but if the 
sum of the given angles is less than 180°, subtract the third 
area from the sum of the others: the result will be double 
the area of the trapezium. 



Demonstration.— Let ABCD (Figs. 156, 157) be 
the trapezium, of which AB, BC, and CD, and the 
angles B and C, are given. 

Join BD, and draw DE and CG perpendicular to 
AB, and CF perpendicular to ED. Then will DCF 
= 180° so (B + C.) Also, draw AH parallel to 
CB, and join DH. 

Then will 2 ABD = AB . DE = AB (EF ± DF) 
= AB.EF±AB.DF = 2 ABC =fc 2 CDH. 



Fig. 156. 




Fig. 157. 



Whence 2 ABCD = 2 BDC + 2 ADB = 2 BCD + 
2 ABC it 2 CDH : the plus sign being used (Fig. 
157) when the sum of the angles is greater than 
180°. 




Sec. IX.] 



CONTENT OF LAND. 



227 



Examples. 

Ex. 1. Given AB = 6.95 chains, BC - 8.37 chains, CD 
= 5.43 chains, ABC = 85° 17', and BCD = 54° 12', to find 
the area of the trapezium. 

As r 0.000000 

: sin. B 85° 17' 9.998527 

AB 6.95 0.841985 

BC 8.37 0.922725 



{ 



: 2ABC 


■ 57.975 


1.763237 


As r 




0.000000 


: sin. 180° 


- (B + C) 40° 31' 


9.812692 


f AB 


6.95 


0.841985 


" 1 CD 


5.43 


0.743800 


: 2CDH 


25.031 


1.398477 


As r 




0.000000 


: sin. C 


54° 12' 


9.909055 


(BC 


8.37 


0.922725 


"(CD 


5.43 


0.734800 


: 2 BCD 


36.862 
57.975 
94.837 
25.031 

2)69.806 


1.566580 




34.903 ch.= 


= 3A.,1R.,38.45P 



Ex. 2. Given AB S. 27° E. 12.47 chains, BC K. 66° E. 
11.43, and CD K 8° W. 9.16 chains, to find the area of 
the trapezium. Ans. 14 A., R., 1.56 P. 

Ex. 3. Given AB S. 45° W. 8.63 chains, BC S. 86° 
30' E. 9.27 chains, and CD K 34° E. 11.23 chains, to find 
the area of the trapezium. 

Ans. 6 A., 2 R., 9 P. 



228 



COMPASS SURVEYING. 



[Chap. V 



360. The above rule is a particular example of a more 
general problem, which may be enunciated thus : — 

Let A, B, C, D, &c. be the sides of any polygon, and let 
the angle contained between the directions of any two 
sides, as B and D, be designated [BD]. Then, leaving out 
any side, we shall have the double area equal to the sum 
of the products of all the other pairs into the sine of their 
included angle. Thus, if the figure were a pentagon, we 
should have 2 the area = BC sin. [BC] + BD sin. [BD] + 
BE sin. [BE] + CD sin. [CD] + CE sin. [CE] + DE sin. 
[DE]. 

Observing that any product must be taken negative, if 
the angle is turned in a contrary direction from the general 
convexity of the figure with reference to the side A. 

Thus, in Eig. 156, we have 2ABCD=AB.BC sin. 
[AB . BC] + BC . CD sin. [BC . CD] - AB . CD sin. [AB . 
CD], the lines BA and CD meeting so as to make the 
angle [AB . CD] present its convexity in the opposite 
direction from that of the figure. 

But, in Fig. 157, we have 2 ABCD = AB . BC sin. 
[AB.BC] + BC.CD sin. [BC.CD] + AB . CD sin. 
[AB.CD]. 

In the pentagon (Eig. 158) we shall 
have 

2 Area = B.C. sin. [B.C.] + B.D. sin. 
[B.D.] + B.E.sin.[B.E.] + C.D.sin. 
[C.D.]+ C.E.sin.[C.E.] + D.E.sin. 
[D.E]. 

In Fig. 159 we have 

2 Area = B.C. sin. [B.C.]+ B.D.sin. 
[B.D.]-B.E.sin. [B.E.]+ CD. sin. 
[C.D.]+ C.E.sin. TC.E.1 + D.E. sin. 
[D.E]. 




Fig. 159. 




Sec. IX.] 



CONTENT OF LAND. 



229 



361. Problem 4. — The bearings and distances of the boun- 
daries of a tract of land being given, to determine its area by 
means of the latitudes and departures of the sides. 



Let ABCDEFG- (Fig. 160) Fig. ieo. 

be the plat of a tract, and let N 
HSrS be a meridian anywhere a 
on the map. Through the 
corners draw the perpendicu- *\ 
lars Aa, Bb, &c. Then, it is evi- 
dent that ABCDEFG = AagQ 
+ Qgf¥ + DdeE - AabB - 
BbcC - GcdD - EefF. 

Now, these various figures 
being trapezoids, their areas 
will be found by multiplying 
their perpendiculars by the 
half-sums of their parallel sides. 

The perpendiculars are the differences of latitude of the 
sides of the tract. The sums of their parallel sides may 
be found as follows : — 

The position of the line N"S being arbitrary, the sum Aa 
+ Bb, corresponding to the first side AB, may be taken at 
pleasure. Now, if from Aa + Bb we take Ah, the whole 
departure of the two sides AB and BC, we have Bb 4- Cc, 
the sum of the parallel sides of BbcC. Similarly, if to 
Bb + Cc we add z'D, the departure of the two sides BC and 
CD, we have Cc + ~Dd; and so on. The whole may be 
arranged in a tabular form, as below, — 




Sides. 


N. 


S. 
Dl 


E. 

qV 

IE 
mE 


W. 
AJc 
Bp 


E. D. D. 


W. D. D. 


Multipliers. 


N. Areas. 


S. Areas. 


AB 


BA; 




Afc + Go 


Aa + Bb, E. 


2 AakB 




BC 




Afc+Bp 


Bb + Cc. E. 
Cc + Dd, E. 


2 BbcC 




CD 
DE 


Cq 


qV — Bp 




2 CcdD 




qD + lE 




Dd-j-Ee, E. 




2 DdeB 


EF 


Em 






ZE + mF 




Ee + F/. E. 


2Ec/F 




FG 





nG 


En 
Go 


mF — Fn 




F/+ Qg, E. 




2 FfgG 


GA 




En + Go 


Gg + Aa. E. 




2 G^aA 



in which the first column contains the sides, and the next 
four the differences of latitude and the departures.; the 



230 



COMPASS SURVEYING. 



[Chap. V. 



fifth and sixth columns contain the whole departures of 
two consecutive sides. These may be called the double 
departures, and the columns headed, accordingly, E.D.D. 
and W.D.D. These double departures are found thus : 
The first, AA; -f Go, is the sum of the departures of GA and 
AB, and is placed in the column of west double departures, 
because both departures are westerly ; the second, AA: + Bp, 
is the sum of those of AB and BC, and is west ; the third 
is Do — Bp, and is east, because D is east of B ; the fourth, 
Dq + El, is east ; and so on. The eighth column contains 
the sums of the parallel sides. These may be called the 
multipliers. They are found by the following process. 
Assuming the first, Aa + B6, at pleasure, designate it 
either east or west. In the figure, the line E"S being to 
the west of AB, the multiplier is east. The double de- 
parture AA: + Bp = Ah being west, subtract it from Aa + 
B6, and we have B6 + Co. To Bo + Co add the next 
double departure, qD — pB = zT>, and we have Cc + Dd; 
qD + IE added to Co + Dd gives Dd + Ee ; IE + mF added 
to Dd-h Ee gives Ee -f F/; mF — En added to Ee + F/ gives 
F/ -f Gg ; and, lastly, En + Go taken from Ef + G# leaves 
G# + Aa. 

The areas are arranged in the last two columns, which 
are headed north areas and south areas for distinction. 
These areas are placed in the above table in the columns 
of the same name as the difference of latitudes of the sides 
to which they belong. 

Had the line NS been drawn so 
as to intersect the plat, some of the 
areas would have been to the west 
of it, and some of the multipliers 
might have been west. Fig. 161 is 
an example of this. 

In this case, we have 

2 ABCDEFG = 2 AabB + 2 BbcC 
+ 2 CcdD - 2Ddr + 2reE -2EefE 
+ 2 EfgQ + 2 Ggs - 2 saA = 2 
AabB + 2BbcC + 2 CcdD - 2 (Ddr 
- reE) - 2 Ee/F + 2 EfgGc + 2 (Qgs - saA.) 




iS 



Sec. IX.] CONTENT OF LAND. 231 

But 2 (Ddr — reE) = Dd . dr — Ee . er = Dd .de — 'Dd . cr — 

Ee . de + Ee . dr ; 

and since Dd : dr : : Ee : er, Dd . er = Ee dr. 

2 (Ddr - reE) = Dd . de - Ee . de = (Dd - Ee) de. 

Whence 2 ABCDEFG = (Aa + B6) ao + (B6 + Cc) 6c + 
(Ce + Dd) ed - (Dd - Ee) de - (Ee + /F) e/ + (/F + G#)/# 
-f- (G^ — Aa) ag. 

The following table exhibits the whole. 



Sides. 


N. 


S. 


E. 

Cz 


W. 
pB 

qC 


E. D. D. 


W. D. D. 


Multipliers. 


N. Areas. 


S. Areas. 


AB 


Ap 




pB+Go 


Bo + Act, W. 




2 Aa&B 


BC 


B? 




pS + qG 


Bo+Cc, W. 




2BoCc 


CD 


Di 




Gi — qG 




Cc + m, w. 

Dd — Ee, W. 




2 CcdD 


DE 




Ei 


Ci + Dt 




2 (DdV — Eer) - 




EF 


Em 




mF 




Di + Fm 




Ee + F/, E. 


2 (Ee/F) 


FG 




Grc 




Go 


Fm — Frc 




F/+G£,E. 




2F/#G 


GA 




A.0 




Fw + Go 


Gg — Aa, E. 




2(Ggs — Aas) 



Here the first multiplier is west, the meridian being to 
the east of the line AB. The subsequent multipliers are 
found as follow :— (B6 + Aa) + (p~B + qC) = B& + Cc ; 
(Bo + Cc) - (0* - qC) = Cc + Dd; (Cc + Dd) - (Gi + Bt) 
= Dd - Ee ; (Bt + Fm) - (Dd - Ee) = (Ee + F/), which 
must be marked east, not only from its position on the 
figure, but also from the fact that the east double departure 
is greater than the west multiplier, which is taken from it ; — 
(Ee + F/) + (Pm-Fn) = F/ + Gg; and (F/ + Gg) — (Fw 
+ Go) = G^ — Aa. 

The areas are arranged so that the additive quantities 
may be in the column of south areas and the subtractive 
in that of north areas. 

From the above investigation the following rule is de- 
rived : — 



Kule. 

Eule a table as in the adjoining examples. Find the cor- 
rected latitudes and departures by Art. 338. Then, if the 
departures of the first and last sides are of the same name, 
add them together, and place their sum opposite the first 
side in the column of double departures of that name ; but 



232 COMPASS SURVEYING. [Chap. V. 

if they are of different names, take their difference and 
place it in the column of the same name as the greater. 
Proceed in the same way with the departures of the first 
and second sides, placing the result opposite the second 
side ; and so on. 

Assume any number for a multiplier for the first side, 
marking it E. for east or W. for west, as may be preferred. 
Then, if this multiplier and the double departure corre- 
sponding to the second side are of the same name, add 
them together, and place the sum with that name in the 
column of multipliers, for a multiplier for that side ; but, 
if the multiplier and double departure be of different 
names, take their difference and mark it with the name 
of the greater, for the next multiplier. Proceed in the 
same manner with the multiplier thus determined and the 
third double departure, to find the multiplier for the third 
side. So continue until all the multipliers have been found. 

Multiply the difference of latitude of each side by the 
corresponding multiplier, for the area corresponding to 
that side. If the multiplier be east, place the product in 
the column of areas which is of the same name as the dif- 
ference of latitude ; but, if the multiplier be west, place 
the product in the column of the opposite name. 

Sum the north and the south areas. Half the difference 
of the sums will be the area of the tract. 

Note. — In working any area, the columns of double departures should 
balance. 

The first multiplier is generally assumed zero. One multiplication is thus 
avoided. When this is done, the last multiplier will be equal to the first double 
departure, but of a different name. 

Examples. 

Ex. 1. Given the bearings and distances as follow, to find 
the area:— 1. K 56%° W. 15.35 ch. ; 2. K 9° W. 19.51 ch. ; 
3. K 66° E. 14.01 ch. ; 4. S. 39|° E. 13.35 ch. ; 5. K 82|° E. 
12.65 ch.; 6. S. 6f° W. 12.18 ch. ; T. S. 52J°W. 20.95 ch. ; 
to find the area. 



Sw. IX.] 



CONTENT OF LAND. 



233 



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234 



COMPASS SURVEYING. 



[Chap. ? 



Ex. 2. Given the bearings and distances as in the ad- 
joining table, to calculate the area. 





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Sec. IX.] CONTENT OF LAND. 235 

Ex. 3. Given the bearings and distances as follow, to 
calculate the area:— 1. K 27° 15' E. 7.75 ch.; 2. S. 62° 25' 
E. 10.80 ch.; 3. S. 7° 55 f E. 9.50 ch.; 4. S. 47° 25' E. 9.37 
ch. ; 5. S. 54° 25' W. 8.42 ch.; 6. K 37° 35' W. 23.69 ch. 

Ans. 22 A., 1 E., 26.17 P. 

Ex. 4. Calculate the area from the following notes: — 

1. K 46° 40' W. 18.41 ch. ; 2. K 54° 30' E. 13.45 ch. ; 3. K 
5° 30' W. 12.65 ch.; 4. S. 74° 55 f E. 17.58 ch; 5. S. 47° 
50' E. 15.86 ch. ; 6. S. 47° 25' W. 16.36 ch. ; 7. S. 62° 35' 
W. 14.69 ch. 

Area, 66 A., 2 E., 21 P. 

Ex. 5. Given the bearings and distances of the sides of a 
tract of land, as follow,— viz. : 1. JST. 43° 25' W. 28.43 ch. ; 

2. K 29° 48' E. 30.55 ch.; 3. S. 80° E. 28.74 ch.; 4. K 
89° 55' E. 40 ch. ; 5. S. 10° 13' E. 23.70 ch. ; 6. S. 63° 55' 
W. 25.18 ch. ; 7. K 63° 45' W. 20.82 ch. ; 8. S. 57° 25' W. 
31.70 ch. : to determine the area. 

Area, 262 A., 2 E., 31 P. 

Ex. 6. Calculate the distances of the third and fourth 
sides, and the area of the tract, from the following notes : — 
1. S. 64° 5' W. 11.18 ch. ; 2. K". 49° 45' "W. 12.91 ch. ; 3. K 
35° 20' E., distance unknown; 4. S. 82° 25' E., distance 
unknown; 5. K 87° E. 13.82 ch. ; 6. K 49° 30' E. 4.95 ch. ; 
7. S. 33° 25' E. 10.80 ch. ; 8. S. 0° 55' E. 9.22 ch. ; 9. S. 
79° 10' W. 14.30 ch. ; 10. H". 52° 15' W. 8.03 ch. 
Ans. 3d side, 12.13 ch. ; 4th, 9.71 ch. ; Area, 57 A., 1 E., 12 P. 

Ex. 7. One corner of a tract of land being in a swamp, 
but visible from the adjacent corners, I took the bearings 
and distances as follow:— 1. S. 45° E. 13.65 ch.; 2. K 38|° 
E. 17.28 ch. ; 3. K 19° W. 23.43 ch. ; 4. S. 58° W. 14 ch. ; 
5. K 87° W. 8.14 ch. ; 6. K 45J° ~W. 9.23 ch. ; 7. S. 28J° W. 
14.60 ch. ; 8. S. If ° E. ; 9. K 79J°E. Eequired the distances 
of the last two sides and the area of the tract. 
Ans. 8th side, 16.44 ch. ; 9th, 20.51 ch. ; Area, 92 A., 1 E., 7 P. 

362. Offsets. If any of the sides border on a water- 
course, or are very irregular, stationary lines may be run as 



236 



COMPASS SURVEYING. 



[Chap. V. 



near the boundary as possible, and offsets be taken as 
directed in chain surveying. The area within the stationary 
lines may then be calculated as above. That of the spaces 
included between those lines and the true boundary is to 
be calculated as in Art. 256. These areas added to or 
subtracted from the former, according as the stationary 
lines are within or without the tract, will give the content 
required. 

"When the tract bounds on a stream, it is usual to con- 
sider the boundary as the middle of the stream, except in 
tide waters or large rivers which are navigable and are thus 
considered public highways. In these cases the boundary 
is low-water mark. 

In reciting the boundaries in title-deeds, the offsets are 
not generally given. The description usually runs thus: 
— Thence S. 43J° E. 10.63 chains to a stone on the bank of 
Ridley Creek, and thence on the same course 1.05 chains 
to the middle of said creek. Thence along the bed of said 
creek, in a southwesterly direction, 37.63 chains; thence 
N". 47° "W., by a marked white-oak on the banks of the 
creek, 25.63 chains to a limestone, corner of John Brown's 
land, &c. 

Examples. 

Ex. 1. Calculate the area from the following field-notes : — 



55 


(4) 
1350 




55 









(3) 


N.26°45'E. 


55 


(3) 
2160 




270 


1929 




396 


1408 




310 


1015 




340 


610 




50 









(2) 


N.56°30'E. 




3050 


Mid. of do. 




3000 


(2)on r.bank 




(1) 


N.36°30'W. 



60 


(6) 
1471 




95 


930 




140 


485 




60 









(5) 


S.51°30'E. 


60 


(5) 
1072 




130 


750 




85 


390 




55 









(4) 


S.84°45'E. 





(1) 
4316 




Middle 


75 

of river. 


PL 

S.45°15'W. 


75 


V) 

826 




100 


420 




60 









(6) 


S.11°45'E. 



Sec. IX. J 



CONTENT OF LAND. 



237 



Sta. 

i~r~ 

\~ 

3 
~T~ 

5 
6 

7 


Bearings. 


Dist. 


N. 


S. 


E. 


W. 


E. D. D. 


W.D.D. 


Mult. 


N.Areas. 


S. Areas. 


N.36>^W. 


30.00 


24.12 




17.84 




47.96 


.00E. 







N.56%E. 


21.60 


11.92 




18.01 




.17 




.17E. 


2.0264 


N.26%E. 


13.50 


12.06 




6.08 




24.09 




24.26E. 


292.5756 




S. 84% E. 


10.72 




.98 


10.68 




16.76 




41.02E. 




40.1996 


S. 51% E. 


14.71 




9.16 


11.51 




22.19 




63.21E. 




579.0036 


S.ll^E. 


8.26 




8.09 


1.68 




13.19 




76.40E. 




618.0760 


s. 4514 w. 


42.41 




29.87 




30.12 




28.44 


47.96E. 




1432.5652 



48.10 48.10 47.96 47.96 76.40 76.40 



294.6020 2669.S444 
294.6020 



Area of offsets calculated as in 
Ex. 1, Art. 257. 



128 A., 2 R., 14.76 P. 



2 )2375.2424 
1187.6212 
= 98.S0145 

= 128.592265 



Ex. 2. Given the field-notes as below of a rneadow 
bounding on a small brook, to calculate the area: — 






(2) 
1132 


55 


1054 


72 


896 


97 


739 


75 


480 


On brook. 







(3) 






1740 




63 


1414 




35 


1237 




87 


1016 




§ 45 


824 




1 50 


652 




si 


551 







452 


75 




295 


75 












**> 






-(2) 







(l) 
1450 






(5) 






(5) 
1344 




<D 

s 


A*' 

1*) 




i 


(4) 
1396 






'(3) 





Ans. 34 A., 3 E., 0.6 P. 

Ex. 3. Required the area of the meadow bordering on a 
mill-race, of which the boundaries are contained in the fol- 
lowing field-notes, the angles given being the deflections 
from the last course : — 





(2) 






(3) 






2.40 


to race-bank. 


c5 

T3 


11.28 




t3 


21.65 






1.96 


( 4 ) 


0Q 


(1) 


S.53PKKW. 


OQ 


(2) 


1-97° 03' 


m 


(3) 


1-97° 45'. 



238 



COMPASS SURVEYING. 



[Chap. V. 



W 






32 


(6) 
9.89 




30 


5.50 




132 


3.00 




40 


1.08 




35 







30° 12' "I 


(5) 




35 


(5) 
1.05 




44 


.11 






(4) 


r8i°i4' 






e>3 





(1) 

9.12 


1- 98° 34' 




(7) 


f- 27° 46' 




(7) 
2.40 




corner 14 


2.26 






2.00 







1.75 


6 




1.50 




32 







12° 14' -J 


(6) 





In calculating the area, it will be necessary first to calcuS 
late the bearings from the observed angles. 

. v 4rea, 15 A., 2 R., 11.5 P v 



Fig. 162. 



F B 



363. Inaccessible Areas. When it is desired 4;o de- 
termine the area of a tract of difficult access, such as a 
pond, a thick copse, or a swamp, it should be surrounded 
by a system of lines as near the boundaries as they can be 
run without multiplying the number of sides unnecessarily. 
Offsets should then be taken to different points of the 
boundary, so as to determine its sinuosities. The areas of 
the parts determined by these offsets, taken from the area 
enclosed in the base lines, will leave the content required. 

Where two base lines make 
an angle with each other, the 
first offset on each should be 
taken to the same point in the 
irregular boundary. Thus, if 
AB and BC (Fig. 162) are two 
adjacent baselines enclosing 
an irregular boundary HDI, the 
first offsets should be. taken at F and E, so situated that the 
offsets FD and ED should meet at the same point D of the 
boundary. The triangular spaces BDF and BDE will then 
be included with the areas belonging to the lines AB and 
BC respectively. 




Sec. IX.] 



CONTENT OF LAND. 



239 



The following examples of the field-notes and calculation 
for the area of a pond will illustrate this subject: — 

Fig. 163 is a plat of Ex. 1 on a scale of 1 inch to 10 chains. 

Fig. 163. 







(5) 




1866 


155 


1805 


25 


1675 





1475 


10 


1250 


55 


950 


22 


800 


75 


475 


r78°55' 







(4) 





115 
55 

90 
105 

22 



42 

42 

r23°51' 






(1) 

1140 


i- f 52° 52' on 

1 1(1). (2). 




1100 


90 




875 


10 




750 


10 




500 


60 




250 


112 


>1' 


75 


112 




(6) 


T 56° 35' 




(6) 






920 






870 


122 




750 


32 




575 


17 




300 


73 




85 


97 


V 


(5) 


f 69° 39' 



240 



COMPASS SURVEYING. 



[Chap. V. 



1 

Sta, 
1 

~2~ 

3 
4 
5 
6 


Bearings. 


Dist. 


N. 


S. 


E. 


W. 


E.D.D. 


W.D.D.j 

29.88 J 


Multipli'r. 


S. Areas. 


N.88°35'W. 


22.80 


.55 






22.78 
~2i3~ 




.00 E. 




N. 9° 40' W. 


13.85 


13.65 








25.11 


25.11 W. 


342.7515 


N. 68° 28' E. 


11.52 


4.23 




10.72 


8.39 




16.72 W. 
12.64 E. 


70.7256 


S. 87° 41' E. 


18.66 




.76 


18.64 




29.36 




9.6064 


S. 18° 2' E. 


9.20 




8.75 


2.85 




21.49 




34.13 E. 


298.6375 


S. 38°33'W. 


11.40 




8.92 




7.10 




4.25 


29.88 E. 


266.5296 



18.43 18.43 32.21 32.21 
Content within the hase-lines, 



59.24 59.24 



2)988.2506 
494.1253 ch. 



Base. 


Dist, 


Offsets. 


Inter. 
Dist. 


Sum of 
Offsets. 


Areas. 




0.00 












0.55 


.82 


.55 


.82 


.4510 




3.12 


.55 


2.57 


1.37 


3.5209 




5.55 


.10 


2.43 


.65 


1.5795 




7.05 


.32 


1.50 


.42 


.6300 




10.00 


2.20 


2.95 


2.52 


7.4340 




12.40 


2.91 


2.40 


5.11 


12.2640 


(1)(2) 


14.80 


1.75 


2.40 


4.66 


11.1840 




17.70 


.33 


2.90 


2.08 


6.0320 




20.15 


.07 


2.45 


.40 


.9800 




22.15 


.60 


2.00 


.67 


1.3400 




22.80 





.65 


.60 


.3900 








45.8054 



.47 


.75 


.47 


.75 


.3525 




1.55 


.22 


1.08 


.97 


1.0476 




4.30 


.55 


2.75 


.77 


2.1175 


(2) (3) 


7.75 


.10 


3.45 


.65 


2.2425 




9.75 





2.00 


.10 


.2000 




11.25 


.25 


1.50 


.25 


.3750 




12.95 


1.55 


1.70 


1.80 


3.0600 




13.85 





.90 


1.55 


1.3950 
10.7901 













1.32 


1.20 


1.32 


1.20 


1.5840 




2.50 


.70 


1.18 


1.90 


2.2420 


(3) (4) 


5.25 


11 


2.75 


81 


2.2275 




7.75 





2.50 


11 


.2750 




9.50 





1.75 





.0000 




11.35 


42 


1.85 


42 


.7770 




11.52 





17 


42 


.0714 



7.1769 



Sec. IX.] 



CONTENT OF LAND. 



241 



Base. 


Dist. 


Offset. 


Inter. 
Dist. 


Sum of 
Offset. 


Areas. 

3.9900 
1.3650 

.3300 
3.8100 
4.3875 
2.9000 
2.2100 

.7015 


(4) (5) 

(5) (6) 
(6)(1) 


.00 

4.75 

8.00 

9.50 

12.50 

14.75 

16.75 

18.05 

18.66 


.42 
.42 
.00 
.22 

1.05 
.90 
M 

1.15 
.00 


4.75 
3.25 
1.50 
3.00 

2.25 

2.00 

1.30 

.61 


.84 

.42 
.22 
1.27 
1.95 
1.45 
1.70 
1.15 










19.6940 


.00 
.85 
3.00 
5.75 
7.50 
8.70 
9.20 


.97 
.73 
.17 
.32 
1.22 
.00 


.85 
2.15 
2.75 
1.75 
1.20 

.50 


.97 

1.70 

.90 

.49 

1.54 

1.22 


.8245 
3.6550 
2.4750 

.8575 
1.8480 

.6100 










10.2700 


.00 

.75 

2.50 

5.00 

7.50 

8.75 

11.00 

11.40 


1.12 
1.12 

.60 
.10 
.10 

.90 
.00 


.75 
1.75 
2.50 
2.50 
1.25 
2.25 

.40 


1.12 
2.24 

1.82 
.70 
.20 

1.00 
.90 


.8400 
3.9200 
4.5500 
1.7500 

.2500 

2.2500 

.3600 



13.9200 



Area within base lines, A. 49.41253 


Double area 


, cut off by 


(1) (2) 


4.58054 


(2) (3) 


1.07901 


(3) (4) 


.71769 


(4) (5) 


1.96940 


(5) (6) 


1.02700 


(6) (1) 


1.39200 




i of 10.76564 = 5.38282 



Area of pond, 



44.02971 = 44 A., OR., 4.75 P. 

16 



242 



COMPASS SURVEYING. 



[Chap. V. 



The following are the field-notes taken for the survey of 
a pond. The area is required. Fig. 164 is the plat, to a 
scale of 1 inch to 10 chains : — 



Fig. 164. 







(1460 ) 


AB 




580 






560 


70 




300 


15 




150 


20 


48' -| 


Sta.F 






Sta.F 






627 







475 


65 




250 







20 


90 


onAB 


Q 627) 


f 44° 5' 





Sta. A 


r f 87° l' 

1 l on AB. 




950 






900 


20 




750 


70 




500 







400 







300 


30 




150 


25 




27 


70 




Sta.E 


[-70° 29'. 



Area, 24 A., 3 R., 20 P. 



Sec. IX.] CONTENT OF LAND. 

364. Compass Surveying by Triangulation. 



243 



When the tract is bounded by straight lines, the area 
may be fonnd by determining the position of each of 
the angular points with reference to one or more base 
lines properly chosen. 

To do this, measure a base from the ends of which all 
the corners of the tract can be seen, and take their angles 
of position. There will thus be a system of triangles 
formed, giving data for calcu- 
lating the content of the tract. 
Thus, if ABODE (Fig. 165) re- 
present a field, measure a base 
FG, and from F and G take the 
bearings, or the angles of posi- 
tion, of A, B, C, D, and E. Cal- 
culate FA, FB, FC, FD, FE, 
and thence the areas of the tri- 
angles FAB, FBC, FCD, FDE, 
and FEA. 

Then, ABODE = FBC + FCD + FDE - FEA - FAB. 




Example. 

To determine the area of a field ABODE, I mea- 
sured a base line FG of 12.25 chains, and at F and G I 
took the angles of position, as follow: — GFA = 63° 15', 
27° 33', GFC = 35° 35', GFD = 58° 25', GFE = 
FGB = 58° 30', FGC = 97° 12', 
FGD = 72° 28', and FGE = 37° 32'. Fig. 165 is a plat 
of this tract, on a scale of 1 inch to 10 chains. 



GFB 

92° 10', FGA = 26° 5', 





Calculation. 






1. To find FA. 




As sin. FAG 


90° 40' 


.000029 


: sin. FGA 


26° 5' 


9.643135 


: : FG 


12.25 


1.088136 


: FA 




0.731300 



244 COMPASS SURVEYING. [Chap. V. 

To find FB. 

As sin. FBG 93° 57' .001033 

: sin. BGF 58° 30' 9.930766 

: : FG 1.088136 



: FB 


To find FC. 


1.019935 


As sin. FCG 


47° 13' 


0.134347 


: sin. FGC 


97° 12' 


9.996562 


:: FG . 




1.088136 


: FC 


1.219045 




To find FD. 




As sin. FDG 


49° 7' 


0.121453 


: sin. FGD 


72° 28' 


9.979340 


:: FG 




1.088136 


: FD 


1.188929 




To find FE. 




As sin. FEG 


50° 18' 


0.113848 


: sin. FGE 


37° 32' 


9.784776 


:: FG 




1.088136 


: FE 


0.986760 


• 


To find 2 FAB. 




sin. AFB 


35° 42' 


9.766072 


FA 




0.731300 


FB 


32.9084 


1.019935 


2 FAB 


1.517307 




To find 2 FBC. 




sinBFC 


8° 2' 


9.145349 


BF 




1.019935 


FC 




1.219045 



2 FBC 24.2286 1.384329 



Sec. IX.] CONTENT OF LAND. 245 





To find 2 FCD. 




sin. CFD 


22° 50' 


9.588890 


CF 




1.219045 


FD 




1.188929 


2FCD 


99.2805 
To find 2 FDE. 


1.996864 


sin. DFE 


33° 45' 


9.744739 


DF 




1.188929 


FE 




0.986760 


2FDE 


83.2585 
To find 2 FEA. 


1.920428 


sin. AFE 


28° 55' 


9.684430 


FE 




0.986760 


FA 




0.731300 


2FEA 


25.2633 


1.402490 


2FBC 


/ 


24.2286 


2FCD 




99.2805 


2FDE 




83.2585 
206.7676 


2 FAB 


32.9084 




2FAE 


25.2633 ' 


58.1717 

2)148.5959 

74.29795 sq.ch 



= 7 A., 1 K., 28.76 P. 

365. If no two points can be found from which all the 
corners can be seen, several points may be taken, and these 
all being connected by a system of triangles with a single 
measured base, or with several if suitable ground for mea- 
suring them can be found, the area may then be calculated. 



246 



COMPASS SURVEYING. 



[Chap. V. 



Thus, (Fig. 166,) if 
ABCDEFG represent a 
tract, and H, I, and K, 
three points such that, 
from H, B, C, D, and E, 
can be seen. From I, all 
the corners can be seen, 
and from K we can see A, h 
G, F, and E. If the angles 
of position of the corners 
relatively to the base lines 
HI and HK be taken, the 
distances IA, IB, IC, ID, 1 

&c. may be found, and thence the areas of IAB, IBC, 
ICD, &c. 

Consequently, ABCDEFG = ICD + IDE + IEF + 
IFG - IGA — IAB - IBC becomes known. 




366. The same principle may be applied to surveying a 
farm by means of one or more base lines within the tract. 
If such lines be run, and the corners be connected by triangles 
with given stations in these bases, the tract may be platted 
and the area calculated. 

In all cases of the application of this method, care should 
be taken to have the triangles as nearly equilateral as possi- 
ble. If any of the angles are very acute or very obtuse, 
the amount of error from any mistake in measuring the 
base or in taking the angles is much increased. 



CHAPTER VI. 

TRIANGULAR SURVEYING. 



367. The method pursued in the last few articles of 
Chap. V. constitutes what is called triangular surveying. It 
consists in connecting prominent points with one or more 
base lines by means of a system of triangles, — the sides of 
these forming bases for other systems until the whole tract 
is covered. 

The positions of these points having thus been accurately 
determined, the minuter configurations may be filled up by 
means of secondary triangles, or by any of the other methods 
of surveying of which we have already treated. 

368. Base. In triangular surveying there is generally 
but a single base measured as a foundation for the work. 
This therefore requires to be measured with extreme care, 
since an error will vitiate the whole work. The precautions 
to insure extreme accuracy are such as almost to preclude 
the possibility of an error. Delambre, in speaking of a 
measurement of this kind in France, says, — 

"To give some idea of the precision of the methods 
employed, it is sufficient to relate the following occurrence 
during the measurement of the base near Perpignan: — One 
day, a violent wind seemed every moment about to derange 
our rules, by slipping them on their supports. After having 
struggled a long time against these difficulties, we finally 
abandoned the work. Three days after, on a calm day, we 
recommenced the work of that whole day, and we only 
found a fourth of a line [one-twelfth of a French inch] dif- 

247 



248 TRIANGULAR SURVEYING. [Chap. VI. 

ference between two measurements, with the one of which 
we were entirely satisfied, but of which the other was 
esteemed so doubtful that we considered it necessary to 
perform the whole work anew." 

369. Reduction to the Level of the Sea. The base 
should if possible be measured on level ground. A smooth 
beach, if one can be found of sufficient length, affords one 
of the best locations. The work then requires no further 
reduction. If the ground is considerably elevated, the 
length must be reduced to what it would have been if the 
same arc of a great circle had been measured on the sea- 
level. This will be shorter than the measured arc in the 
ratio of the radius of the circle of which the measured arc 
forms part to that of the earth. Thus, suppose the arc was 
on ground elevated 300 feet, and a base of 5000 yards had 
been measured: then say, As 3956 miles + 300 feet : 3956 
miles : : 5000 yards : the length required. 

The radius used should be that belonging to the latitude 
in which the work was performed, it being different in dif- 
ferent latitudes in consequence of the oblateness of the earth. 

370. Signals. The base having been measured, the next 
object is to place signals on prominent points over the coun- 
try. Any prominent object may be selected for this pur- 
pose. A tree on a hill, provided it stands so that its trunk 
is visible projected against the sky, the spire of a church 
or any other object so elevated as to be seen from a great 
distance, may be employed. It is in general best, however, 
to employ signals constructed expressly for the purpose. 
Perhaps one of the best is a tall mast with a flag floating 
from the top. The flag waving in the wind can frequently 
be seen when a still object would not attract the attention. 
The observation must, however, be taken to the centre of 
the mast, and not to the flag. The Drummond light, reflected 
in the proper direction by a parabolic mirror, is the best of 
all. Such a signal maybe seen at the distance of sixty miles. 

371. Triangulation. The signals having been placed, 



Sec. IX.] TRIANGULAR SURVEYING. 249 

their relative position should then be determined by de- 
termining their angles of position with each other. In this 
triangulation it is very important to have all the triangles 
as nearly equilateral as possible. It is not always possible 
to obtain triangles so "well conditioned" as would be de- 
sirable. The rule should, however, be strictly observed 
never to employ a triangle with a very acute angle opposite 
to the known side, as a very small error in one of the 
adjacent angles will then produce a very sensible error in 
the calculated distance. For example, suppose the base 
AB were 500 yards long and the adjacent angles were A = 
88° 39' 15" and B = 88° IV 45"; the third angle C would 
then be 3° 3'. 

The calculated distance of AC would be 9394.6 yards: 
an error of 5" in one of the observed angles would cause 
an error in this result of half a yard, — a quantity utterly in- 
admissible in operations of this nature. 

The base generally being short, Fig# 167# 

compared to the sides of the tri- 
angles which it is desirable to 
employ, these should be gradually 
enlarged, without allowing any of 
them to become " ill conditioned." 
The mode in which this is done 
may be seen from Fig. 167, in 
which AB is the base, on which 
two triangles ABC and ABD, both 
well conditioned, are founded. 
The line CD joining their vertices, becomes the base for 
two other triangles DCE and DCF, by means of which the 
line EF may be found. 

The angles at all the points of the triangle should be 
measured. The sum of these should be 180°. If it differs 
but little, a few seconds, from this, the error should be dis- 
tributed among the angles, giving one-third to each. If, 
however, a greater number of observations have been made 
at some stations than at others, they should have a pro 
portionally less share of the error. Thus, suppose the 
recorded angle A is the mean of 5 observations, B the mean 




250 TRIANGULAR SURVEYING. [Chap. VI. 

of 3, and C of 2 : T 2 _ = i of the error should be applied to 
A, r 8 o to B, and ^ to C. 

372. Base of Verification. In order to prove the cor- 
rectness of the observations and calculations, some part of 
the work as distant as possible from the base should be con- 
nected with another carefully measured base, — the coinci- 
dence of the measured and calculated distance of which 
will prove the whole work. In a system of triangulation 
carried over the whole of France, a distance of more than 
600 miles, the base of verification was found to be 

by calculation 38406.54 feet long, 

and by measurement 38407.5 

The difference being only .96 feet, 

which was the total error arising from observations on more 
than sixty triangles. In the United States Coast Survey, 
carried on under the supervision of Prof. A. D. Bache, still 
greater accuracy has been obtained. 



CHAPTER VII. 

LAYING OUT AND DIVIDING LAND. 



SECTION I. 

LAYING OUT LAND. 

Problem 1. — To lay out a given area in the form of a square. 

373. Reduce the given area to square perches or square 
chains, and extract the square root. This root will be the 
length of one side. Erect perpendiculars at the ends equal 
to the base, and the thing is done. 

The side of a square acre is 316.23 links = 12.65 poles 
= 69.57 yards. 

Problem 2. — To lay out a given area in the form of a rect- 
angle, one side being given. 

374. Reduce the area to a denomination of the same 
name as the side. Divide the former by the latter, and the 
quotient will be the length of the other side. 

\ t Examples. * 

Ex. 1. Lay out 10 acres in a rectangular form, one side 
being 12 chains long. Required the other side. 

Ans. 8.33 chains. 

Ex. 2. What must be the length of one side of a rect- 
angle, the area being 15 acres and one side 37.95 perches ? 

Ans. 63.24 perches. 
251 



252 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

Problem 3. — To lay out a given area in a rectangular form, 
the adjacent sides to have a given ratio. 

375. Divide the given area expressed in square chains or 
square perches by the product of the numbers expressing 
the ratio. The square root of the quotient multiplied by 
these numbers respectively will give the length of the sides. 



Demonstration. — If mz and nz represent the sides, and A the area, then 

/A 
will mnz* = A. Whence z = / — . 

sjmn 



Examples. 

*Ex. 1. Required to lay out an acre in a rectangular form, 
so that the length shall be to the breadth as 3 to 2. What 
must be the length of the sides ? 

Ans. 3.873 chains and 2.582 chains. 

Ex. 2. It is desired to lay out a field of 10 acres in a rect- 
angular form, so that the sides may be in the ratio of 4 to 5. 
What are these sides ? 

Ans. 8.944 chains and 11.18 chains. 

Problem 4. — To lay out a given area in a rectangular form, 
one side to exceed the other by a given difference. 

376. To the given area add the square of half the given 
difference of the sides. To the square root of the result 
add said half difference for the greater side, and subtract it 
for the less. 

Construction. — Make AE (Fig. 168) equal to the 
given difference of the sides. Erect the perpendicu- 
lar EG equal to the square root of the given area. 
Bisect AE in F, and make FB = FG : then will AB 
be the greater side, and BE the less. 

For (29.6) AB . BE = EG*. 

The rule may be proved thus : FB* = FG a = GE* 
-\- EF a = area -\- square of half the difference of 
the sides. Then, AB = AF + FB, BC = FB — 
FE. 





Fig. 


168. 




G 






<\ 




4 


' I \ 




/ 


, N 




/ 




^ 


I 


I 


\ 


1 


| 


s 


1 

1 


i 


\ 
\ 




E 


F 



A 



Sec. L] LAYING OUT LAND. 253 

Examples. 

Ex. 1. It is desired to lay out 10 acres in the form of a 
rectangle, the length to exceed the breadth by 2 chains. 
Ans. Length, 11.05 chains; breadth, 9.05 chains. 

Ex. 2. Eequired to lay out 17 A., 3 B., 16 P. in a rect- 
angular form, so that one side may exceed the other by 50 
perches. Ans. Length 84, and breadth 34 perches. 

Problem 5. — To lay out a given area in the form of a tri- 
angle or parallelogram, the base being given. 

377. Divide the area of the parallelogram, or twice the 
area of the triangle, by the base. At any point of the base 
erect a perpendicular equal to the quotient. The summit 
will be the vertex of the triangle, or the end of a side of 
the parallelogram. 

If through the summit of the perpendicular a parallel to 
the base be drawn, any point in that parallel may be taken 
for the vertex of the triangle. 

Problem 6. — To lay out a given area in the form of a tri- 
angle or parallelogram, one side and the adjacent angle being 

given. 

378. As the rectangle of a given side and sine of the 
given angle is to twice the area of the triangle or the area 
of the parallelogram, so is radius to the other side adjacent 
to that angle. 

Fig. 169. 

Demonstration. — By Art. 357 we have (Fig. 169) ^ c ^ 

r : sin. A : : AB . AC : 2 ABC, or (1.6) r . AB : sin. A [ /\ ~~7 

. AB : : AB . AC : 2 ABC; whence sin. A. AB : 2 ABC ; / \ / 

: : r . AB : AB . AC : : r : AC. '> / \ / 



Examples. 

Ex. 1. Eequired to lay out 43 A., 2 R. in the form of a 
parallelogram, one side AB being 54 chains, and the adja- 
cent angle BAC 63°. 



254 



LAYING OUT AND 


DIVIDING 


LAND. [Chap. VII. 


As AB . sin. 


I sm. A 


54 

63° 


A 


. C. 8.267606 
" 0.050119 


: ABCD 


435 ch. 






2.638489 


: : r 








10.000000 


: AC 


9.04 ch. 






1.956214 



Ex. 2. Required to lay out 3.5 acres in the form of a tri- 
angle, one side being 11.25 chains, and the adjacent angle 
73° 25'. Ans. AC 6.49 chains. 

Ex. 3. Given AB K 85° W. 16.37 chains, BDS. 32£° W., 

to determine its length so that the parallelogram ABCD 
may contain 16 acres. Ans. BD = 10.99 chains. 

Ex. 4. The bearings of two adjacent sides of a tract of 
land being 1ST. 85° E. and S. 23° E., required to lay off 10 
acres by a line running from a point in one side 14.37 chains 
from the angle and falling on the other side. 

Ans. Distance, 14.63 chains. 



Fig. 170. 
c ----. 




379. Lemma.— If ABC (Fig. 170) 
be any triangle, and CD a line in 
any direction from the vertex cut- 
ting AB in D, and if AF be taken 
a mean proportional between AB 
and AD, and FE be drawn parallel 
to DC, the triangle AFE = ABC. 



Demonstration. — Since AD : AF : : AF : AB, we have 

(Cor. 2, 20.6) AD : AB : : ADC : AFE ; 

but (1.6) AD : AB : : ADC : ABC, 
therefore ABC = AFE. 

The above lemma will be found very useful in the con- 
structions required in dividing land. 

The mean proportional AF may be found by describing 
a semicircle on AD, erecting a perpendicular BG-, and 
making AF = AG ; or, if the point A is remote, we may 
draw BH parallel to AC, meeting CD in H ; draw HI per- 
pendicular to CD, cutting the semicircle on CD in I; make 



Sec. I.] LAYING OUT LAND. 255 

CK = CI, and draw KF parallel to CA. Then, since BH 
and FK are parallel to AC, the line AD is divided similarly 
to CD (10.6) ; but CK is a mean proportional between CH 
and CD, therefore AF is a mean proportional between AB 
and AD. 

380. Problem 7. — Two adjacent sides of a tract of land 
being given in direction, to lay off a given area by a line running 
a given course. 

Fig. 171. 

Construction. — Take AD (Fig. 171) 

any convenient length. Erect the per- 

2 Area 
pendicular AE = — — - — . Draw the 
* AD 

parallel EF cutting AF in F. Eun FG 
the given course. Take AB a mean pro- 
portional between AD and AG or = N /AD . AG. Then 
BC parallel to GF will be the division line. 

For, by construction, ADF = the given area, and, by lem- 
ma, ABC = ADF. 

AB may be calculated by the following rule : — 

As the rectangle of the sines of the angles adjacent to 
the required side is to the rectangle of radius and the sine 
of the angle opposite to that side, so is twice the area to be 
cut off to the square of that side. 

The truth of this rule is evident from Art. 358. 

Examples. 

Ex. 1. Given AB S. 63° E. and AC K 47° 15' E., to lay 
off 7 acres by a line BC running due north. Required the 
distance on the first side. 



1 / 


f i 




\ 


i y 


i 




\ 


1 s 


i 

i 




\ 


A 


D 


B 


G 



256 



LAYING OUT AND DIVIDING LAND. 



[Chap. VII. 

Here the angles are A = 69° 45', B = 63°, and C = 47° 15'. 
Whence 



A fsin.A 
I sin. B 


69° 45' 


Ar. 


Co 


. 0.027709 


63° 


u 


a 


0.050119 


( rad. 
\ sin. C 


47° 15' 






10.000000 
9.865887 


: : 2 ABC 


140 chains 






2.146128 


: AB 2 








2)2.089843 


AB 


11.09 






1.044921. 



Ex. 2. Given the bearings of two adjacent sides, taken at 
the same station, K 57° 15' "W. and K 45° 30' E., to deter- 
mine the distance from the angular point of a station on 
the first side from which a line running JT. 77° E. will cut 
off 5 acres. Ans. 8.648 chains. 

Ex. 3. Given AB S. 57° E. and AC S. 5° 16' W., to lay 
off" 12 acres by a line running !N". 75° E. Required the dis- 
tance on the first side. Ans. 18.50 chains. 



381. Problem 8. — The directions of two adjacent sides of a 
tract of land being given, to lay off a given area by a line running 
through a given point. 



Construction. — Divide the given 
area by the perpendicular distance 
from P to AC, (Fig. 172.) Lay off 
AD equal to the quotient. Draw 
PE parallel to AB. Make DF 
perpendicular to AD and equal to 
AE. Lay off FC = DE. Then 
CPB will be the division line. 



Demonstration. — Complete the parallelogram ADHI. 

By construction, APD is half the required area ; and, therefore, AIHD con- 
tains the required area. 

Now, because the triangles PIB, HPK, and CDK are similar, and their homo- 
logous sides IP, DC, and HP are equal to the three sides DF, DC, and CF of 
the right-angled triangle DCF, we shall have (31.6) HPK = PBI-f- CDK. To 




Sec. L] LAYING OUT LAND. 257 

these equals add AIPKD, and we have AIHD = ABC ; whence ABC contains 
the required area. 

If the directions of AB and AC and the position of the point P be given by 
bearings, AC maybe calculated as follows: — In API find PI; also find the 
perpendicular PL. Then AD = area -r- PL. Then in DFC we have DF = PI 
and FC = DE to find DC, which added to AD will give AC. 

L? FC be laid off on both sides, another point C / will be determined, 
through which the line may run. 

Examples. 

Ex. 1. Given .the bearings of AB K 34° W., and of AC 
West, to lay off 18 acres by a line running through a point 
P bearing from A K 41° W. 18.85 chains. 

To find PI. 
As sin. I 56° A. C. 0.081426 

: sin. PAI 7° 9.085894 

: : AP 18.85 1.275311 

: PI 



As rad. 
: sin. PAL 
:: PA 
: PL 
Given area, 

AD 12.65 1.102182; 

whence ED = AD - PI = 12.65 - 2.77 = 9.88. 

To find DC. 

FC + FD = 12.65 1.102182 

FC - FD = 7.11 0.851870 

2) 1.954052 
DC = 9.485 .977026 ; 

therefore AC = AD + DC = 12.65 + 9.485 = 22.135 ch. 

Ex. 2. Given the angle BAC = 85°, to lay off 6 acres 
by a line through a spring the perpendicular distances 

17 



2.77 






0.442631 


To find PL and AD 


A. 


C 


0.000000 


49° 






9.877780 


18.85 






1.275311 
1.153091 


180 ch. 






2.255273 



258 



LAYING OUT AND DIVIDING LAND. 



[Chap. VII. 



of which from AB and AC are 3.25 chains and 7.92 chains 
respectively. Eequired AC. 

Ans. AC = 10.40 chains. 

Ex. 3. A has sold B 3J acres, to be laid off in a corner 
of a field, by a line through a tree bearing [North 5.64 
chains from the angular point. Now, the bearings of the 
sides being ET. 46° 15' E. and N. 42° W., it is required to 
find the distance to the division line, measured on the first 
side. Ans. 11.58 ch. 



382. If the point P were exterior to the angle, the con- 
struction and calculation would be perfectly analogous to 
the preceding. The following is an example : — 



Given the angle A = 60°, 
(Fig. 173,) EAP = 90°, and 
AP = 23.42 chains, to cut 
off 14 A. by a line running 
through P. 



Make AD = 



140 



= 5.98. 




^-.-v 



23.42 
Draw PE parallel to AB. 
Erect the perpendicular DF N \ j 

= AE, and make FC = ED. \ \ / 

Then CB will be the divi- • ^ 

sion-line. 

For, as before, AIHD = the given area; but PTTFT = 
PBI + CKD ; .-. HIBK = CKD, and AIHD = ABC. 

r : tan. 30°: : AP (23.42) : AE = DF = 13.52; 
whence CF = DE = AE + AD = 19.50, 

and DC = N /CF 2 -FD 2 = v/33.02x5.98 = 14.05 ; 

AC = 5.98 + 14.05 = 20.03 chains. 

Problem 9. — Three, adjacent sides of a tract of land being 
given in position, to lay off a given area in a quadrilateral form 
by % line running from the first side to the third. 



Sec. I.] 



LAYING OUT LAND. 



259 



CASE 1. 



383. The division line to be parallel to the second, side. 







Fie. 175. 




Conceive the lines CB and Fig. m. 

DA (Figs. 174, 175) to be pro- 
duced till they meet, and cal- 
culate the area of ABE. Add 
this to the given area if the 
sum of the angles A and B is 
greater than 180°, as in Fig. 
174 ; but if the sum be less, 
as in Fig. 175, subtract 
ABCD from ABE : the re- 
mainder will be the area of 
ECD. 

Then say, As EAB is to 
ECD, so is AB 2 to CD 2 . And, as sin. E is to sine of B, so 
is AB^CDtoAD. 

The following is a neat construction : — 

Produce HB and GA to meet in E. Erect AF perpen- 
dicular to AB, and equal to double the area divided by AB. 
Draw FG parallel to AB, meeting AE in G. Then the tri- 
angle ABG will contain the required area. Take ED a 
mean proportional between EA and EG, or let ED = 
x/EA.EG. Through D draw the division line CD : ABCD 
will contain the required area. For (lemma) ECD = EBG; 
whence ABCD = ABG. 

The calculation is more concisely made by the following 
rule : — 

As the rectangle of the sines of the angles A and B is to 
the rectangle of radius and the sine of E, so is twice the 
given area to the difference between AB 2 and CD 2 . 

This difference, added to AB 2 when the sum of the 
angles A and B is greater than 180°, but subtracted when 
the sum is less, will give CD 2 . 

Then, As sine of E is to the sine of B, so is the difference 
between CD and AB to the distance AD. 



260 LAYING OUT AND DIVIDING LAND. [Chap. VII. 



CD 3 : AB 3 ; 

CD 3 *cAB 3 : AB°; 

CD 3 «nsAB 3 : AB a , 

2 ABCD : CD'cssAB 9 . 



Demonstration. — ECD : EBA 
Whence, by division, ABCD : EBA 
consequently, 2 ABCD : 2 EBA 

and 2 EBA : AB 3 

But (Art. 380) sin. A. sin. B : rad. sin. E : : 2 EBA : AB 3 ; 
whence sin. A. sin. B : rad. sin. E :: 2 ABCD : CD 3 «>»AB 3 . 

Examples. 

Ex. 1. Given— 1. K 62° 15' E. ; 2. K 19° 12' W. 7.92 
chains ; 3. S. 87° W., to cut off 5 acres by a line parallel 
to the second side. Bequired the length of the division 
line, and the distance on the first side. 

First Method.— To find ABE, (Art.358.) 



( rad. 
I sin. E 




A. 


C. 0.000000 


24° 45' 


a 


" 0.378139 


( sin. A 


98° 33' 




9.995146 


1 sin. B 


106° 12' 




9.982404 


..J AB 


7.92 




0.898725 


"1 AB 


- 




0.898725 


: 2 ABE 


142.278 




2.153139 


2 ABCD 


100 






2 ECD 


242.278 






As 2 ABE 


142.278 


A. 


C. 7.846861 


: 2 ECD 


242.278 




2.384314 


: : AB 2 


(7.92 
(7.92 




0.898725 
0.898725 


: CD 2 






2)2.028625 


CD 


10.335 




1.014312 


As sin. E 


24° 45' 


A. 


C. 0.378139 


: sin. B 


106° 12' 




9.982404 


::CD-AB 


2.415 




0.382917 


: AB 


5.539 




0.743460 



Sec. I.] 



LAYING OUT LAND. 



261 





Second Method. 






f sin. A 

AS i ' "R 

(_ sm. B 


98° 33' 


A. 


C. 0.004854 


106° 12' 




0.017596 


J rad. 






10.000000 


1 sin. E 






9.621861 


: : 2 ABCD 


100 ch. 




2.000000 


: CD 2 -AB 2 


44.087 




1.644311 


AB 2 


62.7264 


10.33 




Whence CD = 


s/ 106.8134 = 


5, as before. 



Ex. 2. Given— 1. K 26° 47' W. ; 2. K 63° 13' E. 12.72 
chains ; 3. S. 8° 17' E., to cut off 7 acres by a line parallel 
to the second side. The distance on the first side and the 
length of the division line are required. 

Ans. Division line, 10.72 chains; distance, 5.98 ch. 

Ex. 3. Given the bearing of three sides of a tract of 
land, and the length of the middle one, as follow, — viz. : 1. 
K 15° 30' W. ; 2. K 74° 30' E. 11.60 chains ; 3. S. 45° E. : 
to cut off 12 acres by a line parallel to the second side. 
The division line and distance on the first side are re- 
quired. 

Ans. Division line, 16.44 chains; distance, 8.555 ch. 



;k 



£2 



DJ 



LI 



384. If AD and BC (Fig. 176) are &g.m. 

nearly parallel, the following method may 
be employed with advantage : — 

Divide the area by AB : the quotient 
will give the approximate length of the 
perpendicular AI. Draw FE parallel to 
AB, and AK parallel to BH. In AIK 
and ALF find IK and IF. 

FK = FI ± IK, and FE = AB ± FK. 
If the sum of the angles is greater than 180°, the area cut 
off by EF will be too great by the small triangle AFK = 
FK . AI _ __ AFK FK . AI 



a 



Make IL = 



Then will AL be 



2 FE 2 FE 

the corrected perpendicular : AD may thence be found. 



262 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

Examples. 

Ex. 1. Given GA K 87° W., AB N". 5° W. 14.25 chains, 
and BH S. 89° E., to lay off 10 acres by a line parallel to 
AB. 

Here the angles are A = 98° and B = 84° : AK will 
therefore lie between I and F. 

100 

AI = tttt^ = <-02 chains, nearly. 
14.25 J 

In IAF we have IAF = 8° and IA = 7.02; whence IF = 
.987. 

In IAK we have IAK = 6° and IA = 7.02 ; whence IK = 
.738. 

Whence KF = .25 and EF = 14.50. 

KF.AI - , . 
ilence IL = - -— = .06 chains, 

2EF ' 

and AL = 7.02 - .06 = 6.96 chains ; 

whence AD = 7.03 chains. 

The above method is very convenient for field operations. 
EF may be measured directly on the ground; whence FK is 

known, and IL = - ^^ , as before. 
> 2FE ' 

Ex. 2. Given GA North, AB K 89° E. 7.86 chains, and 
BC S. 1° 30' W., to cut off 10 acres by a line parallel to 
AB. Required the distance of the division line from A. 

Ans. 13.00 ch. 



CASE 2. 

385. By a line running a given course. 



Fig. m. 



Construct, as in last case, 
ABG to contain the given 
area. Draw BL according y\ \ 

to the given course. Take y'' 
ED a mean proportional B '" "aw l 




Sec. L] 



LAYING OUT LAND. 



263 



between 


EL and EG: CD p 






Fig 


178. 


parallel 


to BL will be the ^ 


"-^ ra 








division 


line. Eor, 


by the \ 




»^ B 


•\0 




lemma, 


ECD = 


EBG; ' 


\ 


/ \ 
/ \ 






whence 


ABCD = 


ABG, 


\ J 


i 


1 A 


* *• 


the reqi 


lired area. 




\ / 
\ / 

\/ 

A*^ 


I 

\ 


» 


, N. 




. W 




\i> g 
i 
\ 
• 

'A 

N 



- % a»E 



The calculation may be performed by the finding AE and 
the area of ABE ; whence ECD becomes known. The dis- 
tance ED may then be found by Art. 380 ; or, 

Conceive Wn to be drawn parallel to CD, making 'EWn 
— EAB. Then say, As the rectangle of the sines of the 
angles C and D is to the rectangle of the sines of A and B, 
so is the square of AB to the square of Wn. 

And, As the rectangle of the sines of C and D is to the 
rectangle of radius and sine of E, so is twice the given area 
to a fourth term. 

If the sum of the angles A and B is greater than 180°, 
add these fourth terms together ; but, if the sum of A and 
B is less than 180°, subtract the second fourth term from the 
first : the result will be the square of the division line CD. 

Then, As sine of C is to sine of B, so is AB to a fourth 
term ; take the difference between this fourth term and CD, 
and say, As sine of E is to the sine of C, so is this dif- 
ference to AD. 

Demonstration. — Since EraW = EAB, EW is a mean proportional between 
EA and EL. Whence nW is a mean proportional between AP and BL ; there- 
fore AP . BL = wW a . 

Now, by similar triangles, we have 

sin. L (sin. D) : sin. A : : AB : BL, 
and sin. P (sin. C) : sin. B : : AB : AP. 

Whence (23.6) sin. C . sin. D : sin. A . sin. B : : AB« : AP . BL = wW a ; 
and, by demonstration to last case, 

sin. G . sin. D : rad. sin. E : : 2 rcWDC : CD 3 **rcW 3 . 
Draw AMN parallel to BC. Then, in the triangle ABM, we have 
sin. M (sin. C) : sin. BAM (sin. B) : : AB : BM ; 
and, in AND, we have 

sin. NAD (sin. E) : sin.. N (sin. C) : : DN (CD «cBM) : AD. 



264 



LAYING OUT AND DIVIDING LAND. [Chap. VII. 



Examples. 

Ex. 1. Given— 1. K 62° 15' E. ; 2. K 19° 12' W. 7.92 
chains ; 3. S. 87° "W\, to cut off 5 acres by a line perpen- 
dicular to the first side. Eequired the length of the divi- 
sion line, and its distance from the end of the first side. 





First Method. 








As sin. E 


24° 45' 


Ar. 


Cc 


. 0.378139 


: sin. B 


106° 12' 






9.982404 


:: AB 


7.92 






0.898725 


: EA 


18.166 






1.259268 


AB 








0.898725 


sin. A 


98° 33' 






9.995146 


2 ABE 


142.278 






2.153139 


2ABCD • 


100 








2ECD 


242.278 








Then, (Art. 380,) 










c sin. E 
( sin. D 


24° 45' 


Ar. 


Co. 


0.378139 


90° 


« 


u 


0.000000 


( rad. 








10.000000 


( sin. C 


65° 15' 






9.958154 


: : 2 ECD 


242.278 






2.384314 


: ED 2 






t 


2)2.720607 


ED 


22.93 






1.360303 


AE 


18.17 








AD 


4.76 








As sin. C 


65° 15' 


Ar. 


Co 


. 0.041846 


: sin. E 


24° 45' 






9.621861 


:: ED 








1.360303 


: CD 


10.57 






1.024010 



Seo. L] 



LAYING OUT LAND. 



265 



As 



■■{ 





Second Method. 








sin. C 


65° 15' 


Ar. 


Co. 


0.041846 


sin. D 


90° 


u 


a 


0.000000 


sin. A 


98° 33' 






9.995146 


sin. B 
AB 


106° 12' 
7.92 chains 






9.982404 

0.898725 


AB 


u 






0.898725 


riW 2 


65.5913 






1.816846 



As 



r sin. C 
\ sin. D 



Ar. Co. 0.041846 



a 



a 



0.000000 



f rad. 




10.000000 


( sin. E 


24° 45' 


. 9.621861 


:: 2ABCD 


100 chains 


2.000000 


: CD 2 -riW 2 


46.1006 


1.663707 


nW 2 


65.5913 




CD = 


>/111.6919 = 


10.57. 


As sin. C 


65° 15' 


Ar. Co. 0.041846 


: sin. B 


106° 12' 


9.982404 


:: AB 


7.92 


0.898725 


: BM 


8.375 


0.922975 


CD 


10.57 




DX 


2.195 




As sin. E 


24° 45' 


Ar. Co. 0.378139 


: sin. C 


65° 15' 


9.958154 


:: DF 


2.195 


0.341435 


: AD 


4.76 


0.677728 



266 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

It will be seen from the above that the first method is in 
this case the shorter. It has the advantage, also, of first 
giving the value of AD, which of itself is sufficient to de- 
termine the position of the division line. 

In the second method, if AG and BH are nearly parallel, 
the calculation for CD and DN" should be carried to the 
third decimal figure. 

The construction given for this and the preceding case 
admits of easy application on the ground. 

Run the lines CB and GA to their point of intersection ; 
lay out the perpendicular AF ; run FG parallel to AB and 
BL parallel to the division line. Measure EL and EG, and 

make ED = >/EL . EG. 

Ex. 2. The bearings of three adjacent sides of a tract of 
land are— 1. K 26° 47' W. ; 2. K 63° 13' E. 12.72 chains ; 
3. S. 8° 17' E., to cut off 7 acres by a line running due 
east. The distance on the first side and the length of the 
division line are required. 

Ans. Distance, 3.37; division line, 11.11. 

Ex. 3. The bearings of three adjacent sides of a tract of 
land being— 1. H". 78° 17' E; 2. K 5° 13' E. 15.62 chains; 
and 3. 2sT. 63° 43' W., it is desired to cut off 10 acres by a 
line making equal angles with the first and third sides. 
What is the bearing of the division line, and its distance 
from the end of the first side ? 

Ans. Bearing, N". 7° 17' E. ; distance on first side, 6.316. 

If the first and third sides are nearly parallel, the area of 
ABL may be calculated. This taken from ABCD, or 
added to it, according as BL falls within or without the 
tract, will give the area of BLDC, which may be parted off 
as directed in Art. 384. 



Sec.L] 



LAYING OUT LAND. 



267 



CASE 3. 

386. By a line through a given point. 

Produce CB and DA Fig. 179. 

(Fig. 179) to meet in E, 
and calculate the area 
EAB. Thence ECD is 
found. Proceed as in Art. 
381. Thus, calculate or 
measure the perpendicular 

ECD 

PL Lay off EF = ——. 
j PI 

Draw PK parallel to BE, 
meeting AE in K. Erect the perpendicular FG = EK or 
BP, and make GD = FK. Then will the division line pass 
through D. 

Calculation. 




Determine AE. Then ED = EF + v/FE? - EK 2 , and 
AD = ED - EA. 

Examples. 

Ex. 1. Given DA West, AB K 16° 15' W. 6.30 chains, 
BC IS". 57° E., to cut off 3 acres by a line through a 



P, situated K 25° 30' E. 6.09 chains from the 
corner A. 



spring ., 



To find EA, EAB, and ECD. 



As sin. E 


33° 


: sin. B 


73° 15' 


:: AB 


6.30 


: EA 


11.077 


AB 


6.30 


sin. A 


73° 45' 


2 EAB 


66.994 


2 ABCD 


60. 


2 ECD = 


126.994. 



Ar. Co. 0.263891 
9.981171 
0.799341 
1.044403 
0.799341 
9.982294 
1.826038 



268 LAYING OUT AND DIVIDING LAND. [Chap. VII. 





To find PI and EF. 


As rad. 




Ar. Co. 0.000000 


: sin. PAI 


64° 30' 


9.955488 


:: AP 


6.09 


0.784617 


: PI 


5.497 


0.740105 


ECD 


63.497 


1.802753 



EF 11.552 1.062648 

To find AK, EK, and EF. 



As sin. K 


33° At. Co. 0.263891 


: sin. APK 


31° 30' 9.718085 


:: AP 


6.09 0.784617 


: AK 


5.842 0.766593 


AE 


11.077 


EK = FG = 


5.235 


mce KF = 


GD = EF - EK = 6.317. 




To find FD. • 


GD + GF 


11.552 1.062648 


GD- GF 


1.082 0.034227 




2)1.096875 



FD = 3.535 .548437 

Whence AD = EF + FD — EA = 4.01. 

Ex. 2. The bearings of three adjacent sides of a tract of 
land are as follow,— viz. : DA 1ST. 47° E., AB 1ST. 35° 16' W. 
15.23 chains, and BC S. 36° W., to cut off 15 acres by a 
line running through a spring P 9.22 chains distant from 
the first, and 10.55 chains from the second, side. The dis- 
tance of the division line from the end of the first side is 
required. Ans. 10.82 chains from A. 



Sec. I.] LAYING OUT LAND. 

CASE 4. 

387. By the shortest line. 



Produce the lines CB and DA 
(Fig. 180) to meet in E, and calcu- 
late ABE and AE, whence ECD is 
known. £s~ow, the shortest line cut- 
ting off a given area will make equal 
angles with the sides. Therefore EC 

™ -r. ~™^ EC.ED.sin.E 
= ED. But 2 ECD = — 



269 



Fig. 180. 



1<„ 




E 



ED 2 , sin E 
E 



whence we must have AD = EA c* </ 



E.2ECD 

sin. E 



Or, this case may be constructed and calculated as Case 2 
by drawing BL so as to make the angles EBL andELB equal. 

Ex. 1. Given DA K 86° W., AB K 19° 20' E. 16.75 ch., 
and BC K 63° 30' E., to cut off 15 acres by the shortest 
line. The distance on AD and the bearing of the division 
line are required. 

AD = 13.38; bearing of DC, K 11J° W. 

Problem 10. — To cut off a plat containing a given area from 
a larger tract of any number of sides. 

CASE l. 
388. When the division line is to be drawn from one of the 

angles. 



Find by trial the side EF (Fig. 
181) on which the division line will 
fall, and calculate the area ABCDE : 
subtract this area from that re- 
quired; the remainder will be the 
area of AEGr, which may be laid off 
as in Prob 6, Art. 378. Or, 

The course and distance may be 
calculated directly as follows : — 



Fig. 181. 




270 



LAYING OUT AND DIVIDING LAND. [Chap. VII. 



Change the bearings so that the side on which the division 
line will fall may be a meridian. 

Take out the latitudes and departures. The difference 
between the sums of the eastings and westings will be the 
departure of the division line. 

Find the multipliers, assuming that corresponding to the 
division line to be 0. 

Multiply the known latitudes by the multipliers, and 
place the products in the columns of areas. 

Subtract the difference of the sums of the north and south 
areas from double the required area : the remainder will be 
the area corresponding to the side on which the division 
line will fall. Divide this area by the multiplier: the 
quotient will be the latitude of that side. Place it in its 
proper column. 

Take the difference between the sums of the northings 
and southings : this difference will be the latitude of the 
division line. With this latitude and the departure before 
determined calculate the distance and changed bearing, 
from which the real bearing is readily determined. 

Example. 

Ex. 1. Let the bearings and distances be as follows: — 
1. S. 47£° W. 12.21 ch. ; 2. ff. 49° W. 15.28 ch. ; 3. K 13° E. 
13.18 ch. ; 4. S. 76|° E. 17.95 ch. ; 5. S. 89f ° E., to cut off 
35 acres by a line from the first angle and falling on the last 
side. Required the distance on the last side. 

First Method. 



AB 
BC 
CD 
DE 
EA 


Bearings. 


Dist. 


N. 


S. 


E. 


W. 


E. D. D. 


W.D.D. 


Mult. 


N.Areas 


S. Areas. 


S.47^ W. 


12.21 




8.25 




9.00 




8.88 


0000 






N.49°W. 


15.28 
TF.95~ 


10.02 






11.53 




20.53 


20.53 W. 




205.7106 


N. 13° E. 


12.84 


~~4JL9~ 


2.96 






8.57 


29.10 W. 




373.6440 
92.5296 


S.76^°E. 




17.45 




20.41 




8.69 W. 


36.4111 








(10.42) 


( -12) 




17.57 




8.88 E. 





22.86 22.86 20.53 20.53 37.98 37.98 



2 ABCDE 
2 ABCDEG 
2 AEG 



671.8842 
36.4111 

635.4731 

700 
64.5269 



Sec. L] LAYING OUT LAND. 271 

As diff. lat. EA 10.42 A. C. 8.982132 

: dep. .12 1.079181 

:: rad. 10.000000 



: tan. bear. EA 


S. 


0° 40' E. 


8.061313 


Bear. EF 


S 


. 89° 45' E. 




AEF = 




89° 5' 




As cos. bearing 




0°40' 


A. C. 0.000029 


: rad. 






10.000000 


: : diff. lat. 






1.017868 


: dist. 




10.42 


1.017897 


Then, (Art. 378,) 






' 


AS I sin. AEG 




10.42 


A. C. 8.982103 




89° 5' 


" " 0.000056 


2 AEG 




64.5269 


1.809741 


:: r 






10.000000 



EG 6.19 0.791900 



272 



LAYING OUT AND DIVIDING LAND. [Chap. VII. 






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Sec. LJ 



LAYING OUT LAND. 



£73 



Ex. 2. Given as follows :— 1. K 27j-° W. 5 ch. ; 2. K 58° 
W. 9.53 ch.; 3. K 42|° E. 9.60 ch.; 4. S. 81i° E. 14 ch.; 
5. S. 28 J° E. : to lay off 25 acres by a line from the first 
station. The distance on the fifth side is required. 

Ans. 10.76 ch. 

CASE 2. 

389. The division line to run a given course. 

Proceed as in Case 1 to find the area of the tract to a line 
through the ends of the sides on which the division line 
will fall, and the bearing and distance of the closing line. 
The difference between this area and the area to be laid off 
will be the area of a quadrilateral which may be parted off 
as in Art. 385. 



Examples. 

Ex. 1. The boundaries of a tract of land are as follows, — 
viz. : 1. K 75° E. 13.70 ch. ; 2. K 20J° E. 10.30 ch. ; 3. East 
16.20 ch. ; 4. S. 33J° W. 35.20 ch. ; 5. S. 76° W. 16.00 ch. ; 
6. North 9.00 ch. ; 7. S. 84° W. 11.60 ch. ; 8. K 53J° W. 
11.60 ch.; 9. % 36f ° E. 19.60 ch. ; 10. K 22J E. 14.00 ch. ; 
11. S. 76f ° E. 12.00 ch. ; 12. S. 15° W. 10.85 ch. ; 13. S. 18° 
"W. 10.62 ch. It is required to lay off 35 acres from the 
eastern end of the farm by a line perpendicular to the first 
side. The distance of the division line from the second 
corner is required. 

Fig. 182. 



Fig. 182 is a plat of 
this tract. 




274 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

To find BODE and the bearing and distance of EB. 



Sta. 

BC 
CD 
DE 
EB 


Bearings. 


Disfc. 


N. 


s. 


E. 


W. 


E.D.D. 


W.D.D 


Multipl'r. 


Areas. 


N. 20^° E. 


10.30 


9.65 




3.61 




3.23 




.00 E. 




East. 


16.20 






16.20 




19.81 




19.81 E. 




S. 333^° W. 


35.20 




29.35 




19.43 




3.23 


16.58 E. 


486.6230 






19.70 






.38 




19.81 


3.23 W. 


63.6310. 



29.35 29.35 19.81 19.81 23.04 23.04 



Latitude of EB 19.70 

Departure of EB .38 

Tangent of bearing N. 1° 6' "W. 



23.0 


i 


2)550.2540 
275.1270 


A. 


C. 


8.705534 
1.579784 



Cosine of bearing 
Latitude 
Distance EB 



8.285318 

A. C. 0.000080 
1.294466 



19.70 



1.294546 



Now, AB differing in course from FE by only 1°, the fol- 
lowing is the best method of determining the position of 
the division line OP, which, by the conditions, is to be per- 
pendicular to AB. 

Draw ET perpendicular to AB, and find ET and BT : 
OBEP 





" 2 - 


ET ^t*nj 
To find BT and EF. 


' .,; 


cos. EBT 




76° 6' 


9.380624 


EB 






1.294546 


BT 




4.733 


0.675170 


sin. EBT 






9.987092 


EB 






1.294546 


ET 




19.127 


1.281638 


OBEP =350- 


275.1270 = 74.873 


1.874325 






3.915 


0.592687 


iBT 




2.366 




OB 




6.281 





Sec. L] LAYING OUT LAND. 275 

Ex. 2. The boundaries of a tract of land being as follow, — 
viz.: 1. K 39° E. 12.17 chains; 2. S. 88}° E. 14.83 chains; 
3. K 67J° E. 13.32 chains; 4. S. 27|° E. 16.67 chains ; 5. S. 
57J° W. 21.92 chains ; 6. S. 73° W. 18.23 chains ; 7. S. 52i° 
W. 12.00 chains; 8. K 37° W. 22.72 chains; 9. K 67^° 
E. 18.00 chains, — to cut off 55 acres from the east end by a 
line bearing S. 37° E. Required the position of the point 
at which the line must commence. 

Ans. On the first side, at 9.21 chains from the be- 
ginning. 

Problem 11. — To straighten boundary lines. 

390. It often becomes necessary to straighten crooked 
boundaries between farms, so as to leave the same quantity 
of land in each farm. 

First Method.— -If the tracts Fig. 183. 

are platted, this may be done 
approximately by parallels. 
Thus, suppose BCDE (Fig. 
183) was the common bound- 
ary of two farms, and it is 
agreed by the owners to run 
a straight fence from B to 
fall somewhere on EG. Join 
CE, and draw DK parallel to 
it; then join BK, and draw CL parallel thereto: BL will 
be the line required. In open ground, this work may be 
performed in the field by the transit. 

391. Second Method. — Where the lines are straight, the 
method of latitudes and departures will enable us to run 
the line with accuracy. For it is evident that, if we cal- 
culate the area contained by the boundaries BCDELB, it 
should be 0, since the new line is intended to add to the 
contents of neither farm. The calculation would therefore 
be precisely the same in principle as in Art. 388. 




276 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

Examples. 

Ex. 1. Given BC S. 61° E. 16.50 chains; CD K 53J° E. 
20.05 chains ; DE S. 51° E. 18.42 chains ; EG K 10J E. 

Rule a table as below. Then change the bearing so that 
the side on which the new line will fall shall be a meridian. 
Take out the latitudes and departures : the difference be- 
tween the sums of the eastings and westings will be the 
departure of the new line. Find the double departures 
and the multipliers, assuming that corresponding to the 
first side equal to its double departure: that corresponding 
to the division line will thus be 0. Eind the areas : the 
difference between the north and the south areas will be 
the area corresponding to the side on which the line will 
fall. Divide this area by the multiplier of that side : the 
quotient will be the difference of latitude of that side, 
which, as the changed bearing was north, will also be equal 
to its distance. By balancing the latitudes we may obtain 
the difference of latitude of the new line, and thence calculate 
its distance if desired. 



Sec. I.] 



LAYING OUT LAND. 



277 



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h3 P 



278 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

I Ex. 2. Required to straighten the north boundary of the 
tract the field-notes of which are given Ex. 1, Art. 389, 
the new line to run from a point five chains from the be- 
ginning of the tenth side. The bearing and distance of 
the new line, and the position of the point where it strikes 
the fourth side, are desired. 

Ans. Division line, S. 83° 14' E. 40.41 chains to a point 
3.51 chains from the beginning of the fourth side. 

392. Third Method. — When the old lines do not vary- 
much from the position of the new, and are crooked, it will 
frequently be found most convenient to run a "guess-line," 
and take offsets from this to different points of the bound- 
ary. Then calculate the contents of the parts cut off on 
each side of this line. These, if the assumed line were 
correct, must be equal ; if they are not so, divide the dif- 
ference of the areas by half of the length of the "guess- 
line/' and set the quotient off perpendicular to that line. 
Through the extremity of that perpendicular run a parallel 
to the "guess-line," meeting the side of the tract. The 
division line will run through this point, very nearly, if the 
"guess-line" did not differ much from the true one. If 
greater accuracy is required, the operation may be repeated, 
using the line determined by the first approximation as the 
basis of operations. 

393. Fourth Method. — Run a random line from the start- 
ing point to the side on which the new line will fall, and 
calculate the area contained between this line and the 
original boundaries. Then, by Art. 378, run a new line to 
cut off the same area : this will be the line required. 

Thus, (Ex. 1, Art. 390,) the 
bearing of EG (Fig. 184) being 
K 10}° E: run BA S. 79J 
E. 45.45 chains, falling on GE 
at A, distant .69 chains from 
E. in GE produced. 




Sec. L] 



LAYING OUT LAND. 



279 



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280 



LAYING OUT AND DIVIDING LAND. 



[Chap. VII. 



Problem 12. — To run a new line between two tracts of dif- 
ferent prices, so that the quantities cut off from each may be of 
equal value. 

394. This problem is in general a very complicated one, 
and can be best solved by approximation. Thus, run a 
"guess-line," and calculate the area cut off from each tract. 
If these areas are in the inverse ratio of the prices, the line 
is a correct one; if not, run a new line near this, and 
repeat the calculation: a few judicious trials will locate the 
line correctly. 

395. The following cases admit of simple solutions : — 

CASE 1. 

When the old line is straight, and the new line is to run a given 
course. 



Fig. 185. 



The method of solution will best be shown by an ex- 
ample. 

Let the bearings of the 
lines be LA (Fig. 185) K 
46° 45' E., AE S. 71° 
20' E., 24.10 chains, and 
BM K". 10° 35' E., the 
land to the north of 
AB being estimated at 
$80 per acre, and that to 
the south at $100 per 
acre. It is required to 
run a new division line, 
running due east, so as 
not to alter the value of 
the two tracts. 

Through B and A draw BD and AC parallel to the 
division line, and CF parallel to AB, meeting LA pro- 
duced in F. Take AL = \p AD = { AD, and FI a mean 
proportional between AL and AF. Join LB, and draw FE 




Bec.L] laying out land. 281 

parallel to it, meeting AB in E. Then the division line 
will run through E. 

Demonstration.— AL : FI : : FI : AF; .-. AL : AF : : FI a : AF 2 ; but AB 
= | AL ; .-. AD : AF : : f FP : AF 2 : : f BE 2 : AE* : : BE a : f AE a . 

But AD : AF : : ADB : AFB (1.6) : : ADB : ABC : : BE a : £ AE a ; (A) 
and ABC : BEH :: AB 3 : BE 3 ; 

... (23.5) ADB : BEH : : AB a : f AE a ; 

but ADB : f AEK : : AB> : £ AE a , (Cor. 2, 19.6.) 

.♦.BEH = | AEK. 

The operations in the above construction may readily be 
done on the ground. Thus : 

Eun BD, AC, and CF. Measure AF and AD. Calcu- 
late v/| AD . AF, which call M. Then say, As AF + M 
: AF : : AB : AE. Through E run the division line. 

Calculation. 

To find AD. Say, As sin. ADB (43° 15') : sin. ABD (18° 
40') : : AB (24.10) : AD = 11.26. 

To find AF. Say, As sin. ACB . sin. BAF : sin. BAC . 
sin. ABC : : AB : AF ; 

that is, As sin. 79° 25' . sin. 61° 55' : sin. 18° 40' . sin. 81° 
55' : : 24.10 : AF = 8.81 ; FI = V\ AD . AF = 11.13. 

Then, As AF + FI (19.94) : AF (8.81) : : AB (24.10) 
: AE = 10.64; 

Or, As AF + FI (19.94) : AF (8.81) : : AD (11.26) : AK 
= 4.97. 

CASE 2. 

396. The division line to run through a given point E in AB. 

Let the bearings be as in last case. To run the division 
line through a point E in AB 10.64 chains distant from A. 



282 



LAYING OUT AND DIVIDING LAND. 



[Chap. VIL 



Fig. 186. 




Construction. — Take 
AI (Fig. 186) a third 
proportional to BE and 
AE. Let AK = f AI 
and AL = BE. Draw 
LM parallel to BC, 
cutting AB in E ; and 
KM parallel to AB. 
Make LO = ME. 
Join AO, and draw 
GEH parallel to it. 
Then the thing is done. 

Demonstration. — Conceive BC and AL to meet in P. Then we have 

BE : EA : : EA : AL .-. (Cor. 2, 20.6) BE : AI : : BE 3 : EA 3 , and LA : AK 
: : BE 3 : f EA 3 . 

Again : PB : PC : : PD : PA : : PA : PF : : AD : AF; 

but PB : PC : : LN : LO : : LN : NM : : LA : AK : : BE 3 : f EA 3 ; 

whence AD : AF : : BE 3 : £ EA 3 , which agrees with (A) in the demonstration 
of last case. Then, following the steps of that demonstration, we find BEH = 
£ AEG. 

This, like the last case, may readily be done on the 

AE 2 
ground, thus ; Calculate AI = — — -, and make AK = j AI. 

EB 

Lay off on DA produced AL = BE : run 1EM and KM. 

Lay off LO = EM, and run GEH parallel to AO. 



Calculation. 
„ 5 AE 2 

AK - 4EB = 10 " 51 - 

Then sin. M (81° 55') : sin. AKM (61° 55') : : AK (10.51) 
: EM = 9.37 = LO ; 

and, As LA + LO (22.83) : LA - LO (4.09) : : tan. ^PAJ^_42 

(71° 550 : tan - L0A ~ LA0 = 28° 45'; 

LAO = 71° 55' - 28° 45' = 43° 10'. 
But AF bears E. 46° 45' E. ; hence GH bears K 89° 55' E. 



Bec. I.] laying out land. 283 

CASE 3. 

397. When the starting point is in the line AD. 

Given as before to run the line from a point G in AD 
at 4.97 chains from A. 

Produce DA and BC (Fig. 186) to meet in P. Calcu- 
late AP : let the given ratio f be represented by r : then, 
As sin. P (36° 10') : sin. ABC (81° 55') : : AB (24.10) : AP 
= 40.432. 

r.AG 2 

Put __ =.7636 = A; 

AP 

and M 2 = A . PG = 34.67. 



Lay off GD = } A ± V\ A 2 + M 2 = .382 + 5.900,= 6.282, 
(the lower sign being used when G is between A and P.) 
Then GH parallel to DB will be the division line. 



Demonstration.— Since GD = £ A -f- y/\ A 3 + M 3 , 
we have GD — \ A = ^/\ A* + M 3 , and GD 3 — A . GD == M 3 , 

or GD (GD — A) = A . PG ; whence PG : DG : : DG — A : A, 

/r AG 3 \ 
and composition, PD : DG : : DG : A ( ' ) : : AP . DG : r . AG 8 ; 

whence r . PD . AG 3 = AP . DG 3 , 

and r . AG 3 : DG 3 : : AP : PD : : PC : PB : : PF : PA : : AF : AD, 

or, r . AE 3 : EB 3 : : AF : AD. As this agrees with (A) in the demonstra- 
tion to Case 1, the truth of the work is clear. 

Having found AD, the bearing of DB, which is parallel 

to GH, may be found by calculating the angle ADB ; thus : 

As (AB + AD) 35.352 : (AB - AD) 12.848 : : tan. 

ADB + ABD „ „ ADB - ABD 

30° 57J' : tan. = 12° 17' 55 ,f . 

2 2 2 

Whence the angle ADB is 43° 15' 25", and the bearing of 
DB or GH is S. 89° 59' 35" E. 

The whole of the preceding construction might be made 
geometrically, but some of the lines required would be so 
small that no dependence could be had on the work ; the 
method is therefore omitted. 

If the given point were not on one of the lines, the pro- 
blem becomes very complicated. It may, however, be 
solved by running " guess-lines." 



284 LAYING OUT AND DIVIDING LAND. [Chap. VII. 



SECTION II. 

DIVISION OF LAND. 

Problem 1. — To divide a triangle into two parts having a 
given ratio. 

CASE 1. 

398. By a line through one of the corners. 

Divide the base into two parts having the same ratio as 
the parts into which the triangle is to be divided, and draw 
a line from the point of section to the opposite angle, (1.6), 

Examples. 

Ex. 1. A triangular field ABC contains 10 acres, the base 
AB being 22.50 chains. It is required to cut off 4J acres 
towards the point A by a line CD from the angle C. What 
is the distance AD ? 

Calculation. 

As 10 : 4J : : AB (22.50) : AD = 10.125 chains. 

♦ 

Ex. 2. The area of a triangle ABC is 7 acres, the side 
AC being 15 chains. To determine the distance AD to a 
point in AC, so that the triangle ABD may contain 3 acres. 

Ans. AD = 6.43 chains. 

CASE 2. 

399. By a line through a given point in one of the sides. 

Say, As the whole area is to the area of the part to be 
cut off, so is the rectangle of the sides about the angle 
towards which the required part is to lie, to a fourth 
term. 

This fourth term divided by the given distance will give 
the distance on the other side. 



Sec. II.] 



DIVISION OF LAND. 



285 




Demonstration. — Let ABC (Fig. 187) be the given tri- 
angle, and ADE the part cut off. Then we shall have 
(Art. 357) rad. : sin. A : : AB . AC : 2 ABC, and rad. 
: sin. A : : AD . AE : 2 ADE ; wherefore 2 ABC : 2 ADE 
: : AB . AC : AD . AE, or ABC : ADE : : AB . AC : AD 
. AE. 

Examples. 

Ex. 1. Given the side AB = 25 chains, AC = 20 chains, 
and the distance AD = 12 chains, to find a point E in 
AB, snch that the triangle cut off by DE may be to the 
whole triangle as 2 is to 5. 

Calculation. 

As 5 : 2 : : AB . AC (500) : AD . AE (200) ;' 

200 



whence 



AE 



12 



= 16.66 chains. 



Ex. 2. Given AB = 12.25 chains, AC = 10.42 chains, and 
the area of ABC = 5 A. 3 E. 8 P., to cut off 3 acres to- 
wards the angle A by a line running through a point E in 
AB 8.50 chains from the point A. Required the distance 
on AC. Ans. 7.77 chains. 



Fig. 188. 



CASE 3. 

400. By a line parallel to one of the sides. 

Since the part cut off will be similar to the whole, say, 
As the whole area is to the area to be cut off, so is the 
square of one of the sides to the square of the correspond- 
ing side of the part. 

The problem may be constructed thus : 
Let ABC (Fig. 188) be the given triangle. 
Divide AB in F, so that AF may be to 
FB in the ratio of the parts into which 
the triangle is to be divided. Take AD A 
a mean proportional between AF and AB. Then, DE 
parallel to BC will divide the triangle as required. 

For AFC : FCB : : AF : FB, and (lemma) ADE = AFC ; 
therefore ADE : DECB : : AF : FB, 




286 



LAYING OUT AND DIVIDING LAND. [Chap. VII. 



Examples. 

Ex. 1. The three sides of a triangle are AB = 25 chains, 
AC = 20 chains, and BC = 17 chains, to divide it into two 
parts ADE and DECD, having the ratio of 4 to 3, by a line 
parallel to BC. 

Say, As 7 : 4 : : AB 2 (625) : AD 3 = 357.1428; 
whence AD = 18.90 chains. 

Ex. 2. The three sides of a triangle are AB = 25 chains, 
AC = 20 chains, and BC = 15 chains, to divide it into two 
parts ADE and DECB, which shall be to each other as 2 
to 3, by a line parallel to BC. What is the distance on AC 
to the division line ? Ans. 12.65 chains. 

CASE 4. 



401. By a line running a given course. 

Construction. — Divide AB in G, (Eig. 
189,) so that AG- may be to GB in the 
ratio of the parts of the triangle. 
Run CE according to the given course. 
Take AD a mean proportional be- 
tween AF and AG. Then DE paral- 
lel to CF is the division line. 



Fig. 189. 




d B F x B 



For ACG : CGB : : AG : GB, and, by the lemma, ADE 
= ACG. 

ADE : DECB : : AG : GB. 



Calculation. 



In ACF find AF. Then AD = ^AG . AF ; or say, As 

the rectangle of the sines of D and E is to the rectangle of 
the sines of B and C, so is the square of BC to a fourth 
term. 

Then, if the ratio of the parts is to be as m to ft, m cor- 
responding to the triangular portion, multiply this fourth 
term by m, and divide by m -f n : the quotient will be the 
square of DE. Whence AD is readily found. 



Sec. II.] DIVISION OF LAND. 287 

Demonstration. — Draw xy parallel to CF, making kxy = ABC, and draw 
BR parallel to xy. Then, as was shown in Art. 385, sin. D . sin. E : sin. B 
. sin. C ; : BC 2 : xy% and (Cor. 2, 20.6) Axy : ADE or m + n : m : : xy* : DE* 

Examples. 

Ex. 1. The bearings and distances of the sides of a tri- 
angular plat of ground are AB K 71° E. 17.49 chains, BC 
S. 15° W. 12.66 chains, and CA K 63f° W. 14.78 chains, 
to divide it into two parts ADE and DECB, in the ratio of 
2 to 3, by a line running due north. The distance AD is 
required. 

First Method. 

As 



sin. F 


71° 


A. C. 0.024330 


sin. ACF 


63° 45' 


9.952731 


AC 


14.78 


1.169674 


AF 




1.146735 


AG = f 


AB = 6.996 


0.844850 



2)1.991585 



AD = 9.904 ch. .995792 




Second Method. 




71° 


A. a 0.024330 


63° 45 r 


0.047269 


56° 


9.918574 


78° 45' 


9.991574 


12.66 


1.102434 


a 


1.102434 


153.68 


2.186615 


2 




5)307.36 





DE = V 61.472 = 7.841. 

As sin. A 45° 15' A. C. 0.148628 

: sin. E 63° 45' 9.952731 

: : DE 7.841 0.894371 

: AD 9.902 0.995730 



288 



LAYING OUT AND DIVIDING LAND. 



[Chap. VII. 



Ex. 2. Given AB K 63° W. 12.73 ch., BC S. 10° 15' W. 
8.84 ch., and CA K 77° 15' E. 13.24 ch., to determine the 
distance AD on AB so that DE perpendicular to AB will 
divide the triangle into two equal parts. 

Ans. AD = 8.049 ch. 

CASE 5. 



Fig. 190. 




402. By a line through a given 'point 

Let ABC (Fig. 190) be the tri- 
angle to be divided into two parts 
CLK and ABKL, which shall be 
to each other as the numbers m 
and n : the division line to run 
through a given point P. 



Construction. 



Bisect BC in D ; divide CA in F, so that CF : FA : : m : 
n. Through P draw HPE parallel to BC. Join ED ; draw 
FG parallel to it, and complete the parallelogram CH. 
Make GI perpendicular to BC and equal to EP. With the 
centre I and the radius PH, describe an arc cutting BC in 
K ; then KPL will be the division line. 

If IG is greater than IK, the question is impossible in the 
terms proposed. The triangular part will then be adjacent 
to one of the other angular points, and a construction alto- 
gether analogous to the above will fix the position of the 
division line. 

Demonstration. — Conceive DA, DF, and EG to be joined. Then, since CD = 
\ BC, ADC = \ ABC, and, because CF : FA : : m : n, we have by composition 



CA : CF : : m -f n : m; whence CFD = 



m-\-n 



CAD. But CDF = CEG, andCH 



= 2 CEG .-. CH = CAB, and by demonstration (Art. 381) CKL = CH ; 

m -\-n 



therefore CKL = 



m -f- n 



CAB. 



Sec. II] DIVISION OF LAND. 289 

Calculation. 

Find PE, EC, and FC = AC ; then CE : CF : : CD 

m -f n 



(I BC) : CG, and KG = s/ EI 2 - IG 2 = V PH 2 — PE 2 . 
Finally, CK = CG ± GK. 

Examples. 

Ex. 1. Given the bearings and distances of the adjacent 
sides of a triangular tract,— viz. : CA K 10° 17' W. 13.25 
ch., CB N". 82° 5' W. 13.75 ch.,— to divide it into two por- 
tions ABEL and KLC in the ratio of 4 to 5, by a line through 
a point P 3S". 28 W. 7.85 chains from the corner, C. The 
distance CK is required. 





Calculation. 








To find PE and EC. 






As sin. PEC 


108° 12' 


A. 


C. 0.022289 


: sin. PCE 


17° 43' 




9.483316 


:: PC 


7.85 




0.894870 


: PE 


2.515 




0.400475 


As sin. PEC 


108° 12' 


A. 


C. 0.022289 


: sin. CPE 


54° 5'- 




9.908416 


:: PC 






0.894870 


: CE 


6.692 




0.825575 


- 


To find CG. 






AsCE 


6.692 


A. 


C. 9.174425 


: CF = fCA 


7.361 




0.866937 


: : CD = J CB 


6.875 




0.837273 


: CG = EH 


7.562 




0.878635 


EP 


2.515 






PH = IK = 5.047 






19 







290 



LAYING OUT AND DIVIDING LA 


ND. [Chap. VII. 




To find KG and CK. 




KI + IG 


7.562 


0.878635 


KI-IG 


2.532 


0.403464 

2)1.282099 


KG = 


4.376 


.641049 


CG = 


7.562 




CK = 


11.938 





Ex. 2. Given AB JST. 46° 15' E. 8.80 ch., AC S. 65° 15' E. 
11.87 ch., to determine the distance AK to a point K in AB 
so that a line from K through a spring P K". 80° E. 5.90 ch. 
from A may divide the triangle into two equal parts. 

Ans. AK = 8.58 ch., or 6.244 ch. 



Problem 2. To divide a trapezoid into two parts having a 
given ratio. 

CASE 1. 



Fig. 191. 
D G F 




403. By a line cutting the parallel sides. 

a. Divide DC and AB (Fig. 191) 
in F and E so that the parts may have 
the same ratio as the parts into which 
the trapezoid is to be divided: join 
EF and the thing is done. 

b. If the division line is to pass through a given point G 
in one of the parallel sides. Determine F and E as before ; 
then lay off EH = FG, and GH will be the division line. 

c. If the division line is to pass 
through a point P (Fig. 192) not in 
AB or CD. Determine EF as 
before. Bisect it in I. Through P 
and I draw the division line GH. 

Should GH cut either of the non- a eh b 

parallel sides before it does both of these, one of the por- 
tions will be a triangle. It will then be necessary to calcu- 
late the area of the whole tract, whence that of each por- 
tion is found. Then, by Art. 381, lay off a triangle by a line 
through P so as to contain the required area. 



Fig. 192. 
D G F 


c 


m/...V j L 


\ 


/ \ 


\ 


1/ ; \ 


\ 



Seo. IL] 



DIVISION OF LAND. 



Calculation. 



291 



Through P draw MPL parallel to AB, and from the data 
given find AM and MP. 

Then DA : AM : : AE — DF : AE - LM ; whence LM 
and PL are known. 

J AD : : PL : GF = EH; and DG = 



But AM — J AD 
DF - FG. 



Examples. 



Ex. 1. Given AB E. 9.10 ch., BC K 14° 20' W. 4.40 ch., 
CD W. 6.95 ch., and DA S. 14° W. 4.39 ch., to divide the 
tract into two parts having a ratio of 3 to 4 by a line HG 
through a spring !N". 47° E. 4.40 ch. from the corner A; the 
smaller division to be next to AD. Required the distances 
of the division line from A and D. 





Calculation. 






To find AM and MP. 




As sin. M 


76° 


A. C. 0.013096 


: sin. APM 


43° 


9.833783 


:: AP 


4.40 


0.643453 


: AM 


3.093 


0.490332 


And As sin. M 




A. C. 0.013096 


: sin. PAM 33° 


9.736109 


:: AP 




0.643453 


: PM 


2.470 


0.392658 



To find EH, AH, and DG. 

DF = f DC = 2.979, and AE = f AB = 3.90. 
Then, As AD (4.39) : AM (3.093) : : AE — DF (.921) : A& 
— ML = .649; 

whence ML = 3.251, and PL = 3.251 — 2.470 = .781. 
As AM — | AD (.898) : J AD (2.195) : : PL (.781) : FG = 
EH = 1.909. Finally, AH = AE + EH = 5.81, and DG 
= df — FG = 1.07. 

Ex. 2. Given AB S. 62° 50' E. 14.93 ch., BC K 7° 30' W. 
6.29 ch., CD K 62° 50' W. 11.88 ch., DA S. 21 W. 5.18 ch., 



292 



LAYING OUT AND DIVIDING LAND. 



[Chap. VII. 



to determine DG and AH so that a line joining G and H 
will pass through P K 75° 50' E. 6.20 ch. from A, and cut 
off one-third of the area of the tract towards AD. 

Ans. AH = 3.40 ch. ; DG = 5.53 ch. 

CASE 2. 

404. The division line to be parallel to the parallel sides. 



Fig. 193. 




Let ABCD (Fig. 193) be the trape- 
zoid to be divided into two parts AEFD 
and FEBC having the ratio of two 
numbers m and n by a line EF parallel 
to AD or BC. 



Construction. l 

Join CA, and draw DH parallel to it. Join CH. Divide 
HB in I so that HI : IB : : m : n. Produce CD and BA to 
meet in G, and take GE a mean proportional between GI 
and GB. Join CI, and draw EF parallel to AD : then will 
EF be the division line required. 

Demonstration. — Because DH is parallel to CA, AHC = ADC (37.1) ; .*. 
ABCD = BCH, and, since HB is divided in I so that HI : IB : : m : n, we have 
CHI : CIB : : m : n (1.6.) These triangles are therefore equal to the parts into 
■which the trapezoid is to be divided. But (lemma) GEF = GIC : therefore 
EBCF = ICB, and EF is the division line. 

Calculation. 
EF may be found by the formula EF 2 = ; ; 



then BC svs AD : EF <svs AD : : AB : AE. 



m + n 



Demonstration.— GBC : GAD :: BC a : AD a ; .-. (17.5) ABCD : GAD :: 

BC 2 — AD 3 : AD 3 . 

Similarly, GEF : GAD : : EF 3 : AD 3 .-. (17.5) AEFD : GAD : : FE 3 — AD2 : AD a ; 

whence ABCD : AEFD : : BC 3 — AD 3 : FE 3 — AD 3 ; 

or, m+ n: m:: BC 3 — AD 3 : FE 3 — AD 3 : 

consequently (m + n) FE 3 — m AD 3 — n AD 3 = mBC 4 -m AD 3 ; 

m BC 3 + n AD a 

or, (m 4- n) FE 3 = m BC 3 -f n AD2, and FE 3 = . 

m -f- n 

Again : Draw AKL parallel to DC. Then BL : EK : : AB : AE ; 

or, BC — AD : FE — AD : : AB : AE. 



Sec. II.] DIVISION OF LAND. 293 

Second Method. 
The distance AE may be calculated thus :— 

Find G-A and GD ; thence GC and GB are known : 

then GC : GD : : GA : GH ; whence HB and HI are known,, 



and therefore GE = v/ GI . GB is known. 



Examples. 

Ex. 1. Given AB S. 14° W. 4.39 ch., BC E. 9.10 eh., CD 
K 14° 20' W. 4.40 chains, and DA W. 6.95 chains, to divide 
the trapezoid into two parts AEFD and BEFC in the ratio 
of 2 to 3, by a line EF parallel to the sides BC and DA. 
Kequired the distance AE on the first side. 

__ 2 m . BC 2 + n . AD 2 165.62 + 144.9075 

El! == = 

m + n 5 

310.5275 _, Arr 
= = 62.1055 ; 



whence EF = V 62.1055 = 7.88. 

And BC - AD (2.15) : EF - AD (.93) : : AB (4.39) : AE 
= 1.90. 

Ex. 2. Given AB S. 87° 15' E. 6.47 chains, BC N. 23° 
30' K 10.32 chains, CD S. 64° 45' W. 9.30 chains, and DA 
S. 23° 30' W. 5.55 chains, to determine the distance AE of 
a point E, situated in AB, such that EF parallel to AD 
may divide the trapezoid into two parts AEFD and 
EBCF having the ratio of 4 to 5. 

Ans. AE = 3.36 chains. 



294 LAYING OUT AND DIVIDING LAND. [Chap.VIL 

Problem 3. — To divide a trapezium into two parts having a 
given ratio. 

CASE 1. 

405. The division line to run through a given point in one of 
the sides. 

Let ABCD (Fig. 194) represent ri s- 194 - c 

the trapezium and P the given 
point ; and let m : n represent the 
given ratio. 

Construction. — Determine I, as ^/ /'__ 

in Art. 404. Join PI, and draw G H 

CF parallel to it : then will PF be the division line. 

For if CH and CI be joined, CHD = ABCD ; and, since 
HCI : ICD : : m : n, HCI and ICD will be equal to the two 
parts into which the quadrilateral is to be divided. But, 
since PI is parallel to CF, we have 

GC : GP : : GF : GI; .\ (15.6) GPF = GCI, andPFDC = CID. 

Calculation. 
In GAB find GA and GB. 
Then GC : GB : : GA : GH; 

whence HD and HI become known ; 

and GP : GC : : GI : GF. 

Finally, AF = GF - GA. 

Examples. 

Ex. 1. Given AB K 25}° E. 4.65 chains, BC K 77° E. 
6.30 chains, CD South 7.30 chains, and DA K 78J° W. 
8.35 chains, to divide the trapezium into two equal parts by 
a line EF running through a point P in BC distant 2.50 
chains from B. AF is required. 



Sec. II.] DIVISION OF LAND. 295 







Calculation. 








To find GA and GB 




As sin. 


G 


24° 45' 


A. C. 0.378139 


: sin. 


GBA 


51° 15' 


9.892030 


:: AB 




4.65 


0.667453 


: AG 




8.662 


0.937622 


AD 




8.35 




GD 




17.012 




As sin. 


G 


24° 45' 


A. C. 0.378139 


: sin. 


GAB 


104° 


9.986904 


:: AB 






0.667453 


: BG 




10.777 


1.032496 


BC 




6.30 




GC 




17.077 
To find GH. 




AsGC 




17.077 


A. C. 8.767588 


: GB 




10.777 


1.032496 


:: GA 




8.662 


0.937622 


: GH 




5.466 


0.737706 



HI = i (GD - GH) = 5.773 and GI = GH + HI = 11.239. 

To find GF and AE\ 

As GP 13.277 A. C. 8.876900 

: GC 17.077 1.232412 

: : GI 11.239 1.050727 

: GF 14.456 1.160039 

AG 8.662 

AF 5.794. 

Ex. 2. Given AB K 27J° W. 19.55 chains, BC East 
18.92 chains, CD S. 1J° E. 10.49 chains, and DA S. 56° W. 
12.25 chains, to find BF, so that a line run from a point 



296 



LAYING OUT AND DIVIDING LAND. 



["Chap. VII 



P in AD 6 chains from A may divide the trapezium into 
two parts ABFP and PFCD having the ratio of 5 to 4. 

Ans. BF = 9.00 ch. 

CASE 2. 

406. The division line to run through any point 

Let ABCD (Fig. 195) H* 195. 

foe the given trapezium 
and P the given point. 
Determine I, as in the 
last two articles, and bi- 
sect GI in K. Through 
P draw OPM parallel to 
GD, meeting GB in 0. }/ 

Join KO, and draw CL n 

parallel to it. Through 

L draw LM parallel to GB. Make LK perpendicular to 
AD and equal to OP. "With the centre N and radius 
equal to PM, describe an arc cutting AD in F. Then FPE 
will be the division line. 



g«si: 






H 




~1^M 



Demonstration. — As was proven, Art. 381, GFE = GOML = 2 GOL = 
2 GCK = GCI : whence ABEF = ABCI. But CI divides the trapezium into 
two parts having the given ratio ; therefore, EF does so likewise. 

Calculation. 

Find GB, GA, GH, and GI. Then in OBP find OB and 
OP : thus GO is known. And because GO : GO : : GK : 
GL, GL is known ; but PM == GL - OP. Hence, in LKF 
we have LN and !NTP to find LF. 



Examples. 

Ex. 1. Given AB K 25f ° E. 4.65 chains, BC S". 77° E. 
6.30 chains, GD South 7.30 chains, and DA F. 78J° W. 8.35 
chains, to part off two-fifths of the tract next to AB by a 
line through a spring S. 54f ° E. 2.95 chains from the second 
corner. The distance AF is required. 



Sec. II.] DIVISION OF LAND. 297 

Calculation. 

As in Ex. 1, last case : GB = 10.777, GA = 8.662, GC 
= 17.077, GD = 17.012, GH = 5.466, GI = (GH + § HD) 
= 10.084, and GK = 5.042. 

To find OB and OP. 

As sin. BOP 24° 45' A. C. 0.378139 

: sin. BPO 23° 30' 9.60070O 

: : BP 2.95 0.469822 

: OB 2.81 0.448661 

GB 10.777 

GO 7.967 

As sin. BOP 24° 45' A. C. 0.378139 

: sin. OBP 131° 45' 9.872772 

: : BP 0.469822 

: OP 5.257 0.720733 

To find GL. 

As GO 7.967 9.098705 

: GC 17.077 1.232412 

: : GK 5.042 0.702603 

: GL 10.807 1.033720 

STF = GL - OP = 5.55. 
Whence LF = v/P^IF= 1.779 ; 

whence AF = GL + LF - GA = 3.924. 

Ex. 2. Given AB K 27i° W. 19.55 chains, BC East 
18.92 chains, CD S. 1|° E. 10.49 chains, and DA S. 56° 
W. 12.25 chains, to divide the quadrilateral into two parts 
ABEF and FECD in the ratio of 5 to 4, by a line EF 
through a spring P, which bears from B S. 70J° E. 11.52 
chains. The distance AF is required. 

Ans. AF = 5.01 ch. 



298 



LAYING OUT AND DIVIDING LAND. 



[Chap. VH. 




CASE 3. 

407. The division line to be parallel to one side. 

Let ABCD (Fig. 196) re- Yig.m. 

present the trapezium which 
is to be divided into two 
parts having the ratio of m 
to n by a line parallel to CD. 

Construction. — Deter- 
mine H and I, as in the pre- 
ceding articles. Take GF a mean proportional between 
GI and GD : then EF, parallel to CD, will be the division 
line. 

For, as was demonstrated, (Art. 404,) 

ABCD = HCD, 
CHI : CLD :: m : n. 
GCI = GEF; 
ICD = EFDC, 
and HCI = ABEF: 

whence ADEF : FECD 



and 

But (lemma) 



m : n. 



Fig. 197. 



' 



If the division line 
is to be parallel to the 
shorter side AB (Fig. 
19T.) Draw CK paral- 
lel to AB, and take GF 
a mean proportional 
between GI and GK ; Q ^^_ 
or, join BD, and draw 
CH' parallel to it. Divide AH' in I', so that 

AF : FH' : : m : n, 

and take GF a mean proportional between GA and GF. 
Then will EF, parallel to AB, be the division line. 



h 




Sec. II.] DIVISION OF LAND. 299 

Calculation. 
First Method. — Find, as in the preceding articles, GH 
and GI. Then GF = v/GI.GD, or= </GI.GK. 

Second Method. — Draw xy (Fig. 196) parallel to EF, so as 
to make Gxy = GAB, or Gxy = GCD, (Fig. 197.) Then we 
shall have 

sin. E . sin. F : sin. A . sin. B : : AB 2 : xy 2 , (Fig. 196,) 
or sin. E . sin. F : sin. C . sin. D : : CD 2 : xy 2 ; (Fig. 197 ;) 

yn CD 2 "4" n xv 2 
and (Art. 404) EF 2 = — ' - > (Fig. 196 ;) 

„.„ m . xy 2 -+• n . AB 2 „,. 

or EF 2 = ^— L , (Tig. 197.) 

m + n 

Demonstration. — Draw AM and BN (Fig. 196) parallel to EF. 

Then sin. M . (sin. E) : sin. B : : AB : AM, 

and sin. N . (sin. F) : sin. A : : AB : BN ; 

(23.6) sin. E .sin. F : sin. A . sin. B : : AB a : AM . BN. 

Now, since Gxy = GAB, Gx is a mean proportional between GA and GN. 
Wherefore xy is a mean proportional between AM and BN. Hence, AM . BN 

consequently, sin. E . sin. F : sin. A . sin. B : : AB a : xy\ 

If EF is parallel to AB, (Fig. 197,) the demonstration will be precisely similar 
to the above. 

Examples. 

Ex. 1. Given the bearings and distances as follow, — viz. : 
AB K 25|° E. 4.65 chains, BO N". 77° E. 6.30 chains, CD 
South 7.30 chains, and DA K 78J° W. 8.35 chains,— to 
divide the trapezium into two parts ABEF and FECD, 
having the ratio of 2 to 3, by a line EF parallel to AB. 
AF and EF are required. 

Calculation. 

First Method.— As in Ex. 1 of Art. 405, we find GA = 
8.662, GB = 10.777, GC = 17.077, GD = 17.012, GH - 
5.466, and GI = GH + |HD = 10.084. 



300 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

To find GK and GF. 



AsGB 


10.777 


A. 


C. 8.967504 


: GA 


8.662 




0.937622 


:: GO 


17.077 




1.232412 


: GK 






1.137538 


GI 


10.084 




1.003633 








2)2.141171 


GF = V GI . 


GK = 11.765 


1.070585 


GA = 


8.662 






AF = 


3.103 
To find EF. 






AsGA 


8.662 


A. 


C. 9.062378 


: AB 


4.65 




0.667453 


:: GF 


11.765 




1.070585 


: EF 


6.316 

Second Method. 




1.800416 


f sin. E 

A a J 


128° 45' 


A. 


C. 0.107970 


\ sin. F 


76° ^ 


a 


" 0.013096 


f sin. C 


77° 




9.988724 


\ sin. D 


78° 15' 




9.990803 


J CD 


7.30 




0.863323 


:: [CD 






0.863323 


: xy 2 


67.18 

2 

134.36 




1.827239 


3 AE 2 


64.8675 
5)199.2275 






EF = 


V 39.8455 = 
To find AF. 


6.312. 




As sin. G 


24° 45' 


A. 


C. 0.378139 


: sin. E 


128° 45' 




9.892030 


: : FE - AB 


1.662 




0.220631 


: AF 


3.096 




0.490800 



Sec. II.] DIVISION OF LAND. 301 

Ex. 2. Given the bearings and distances as in Ex. 1, 
to divide the trapezium into two parts AFED and FECB, 
having the ratio of 3 to 2, by a line EF parallel to BO. AF 
and EF are required. 

Ans. AF = 1.60 chains; EF = 7.66 chains. 

Ex. 3. Given as in Ex. 1, to divide the trapezium into 
two parts ABEF and FECD, in the ratio of 2 to 3, by a 
line EF parallel to CD. AF and EF are required. 

Ans. AF = 4.44 chains; EF = 5.62 chains. 

CASE 4. 
408. The division line to run any direction. 

Let ABCD (Fig. 198) be Fig> 198> ' 

the trapezium to be divided 
into two parts ABEF and 
FECD, in the ratio of m to yY ^ 

n, by a line EF running any ^''^\\\ 

course. s*** \\\ 

The construction of this ^___ V-.Y 

case is the same as that of G H axnif k«d 

the last, — CK being drawn so as to be of the same course as 
EF. 

Calculation. 

Conceive xy and vw to be drawn so as to make Gxy = 
GAB, and Gvw = GCD : then will vwyx be equal to ABCD. 
It will also be divided by EF into two parts having the 
ratio of m to n. 

Find xy 2 and vw 2 by the proportions 

sin. E . sin. F : sin. A . sin. B : : AB 2 : xy 2 , 
and sin. E . sin. F : sin. C . sin. D : : CD 2 : vuf, 

the truth of which has been proven in the demonstration to 
rule for Art. 407. 

m . vw 2 + n . xy 2 



Then (Art. 404) EF 2 = 

Dra^ 
and P. 



m + n 
Draw AOP parallel to BC, meeting BIsT and EF in O 



302 LAYING OUT AND DIVIDING LAND. [Chap. VII. 

Then sin. BOA (sin. E) : sin. BAO (sin. B) : : AB : BO, 
and sin. PAF (sin. G) : sin. P (sin. E) : : PF (EF — BO) 
: AF. 

The calculation may otherwise be made by finding GH 
and GI, as in Arts. 406, 407, and also GK. Then GF — 

v'GTTGK. 

Example. 

Ex. 1. The bearings and distances being as in the ex- 
amples in last case, it is required to divide the trapezium 
into two parts ABEF and FECD, having the ratio of 2 to 
3, by a line perpendicular to AD. To find AF and EF. 

Ans. AF = 3.84; EF = 5.76. 



CHAPTER VIII. 

MISCELLANEOUS EXAMPLES. 

Ex. 1. Two sides of a triangle are 32 and 50 parches 
respectively. Required the third side, so that the area 
may be 3 acres. Ans. 31.05 P. or 78 P. 

Ex. 2. A gentleman has a garden in the form of a rect- 
angle, the adjacent sides being 120 and 100 yards respec- 
tively. There is a walk half round the garden, which 
takes up one-eighth of the ground. "What is its width ? 

Ans. 7.05 yards. 

Ex. 3. The three sides of a triangle are in the ratio of 
the numbers 3, 4, and 5. What are their lengths, the area 
being 2 A., 1 R., 24 P.? 

Ans. 6 chains, 8 chains, and 10 chains. 

Ex. 4. The diameter of a circular grass-plat is 150 feet, 

and the area of the walk that surrounds it is one-fourth of 

that of the plat. Required the width. 

Ans. 8.85 feet. 

Ex. 5. To determine the height of a liberty-pole which 
had been inclined by a blast of wind, I measured 75 feet 
from its base, the ground being level, and took the angle 
of elevation of its top 67° 43' 30", the angle of position 
of the base and top being 5° 37'. Then, measuring 100 
feet farther, I found the angle of position of the bottom 
and top to be 2° 29'. Required the length of the pole. 

Ans. 194 feet. 

Ex. 6. The distances from the three corners of a field in 
the form of an equilateral triangle to a well situated within 
it are 5.62 chains, 6.23 chains, and 4.95 chains respectively. 
"What is the area ? Ans. 4 A., R., 6 P. 

303 



304 MISCELLANEOUS EXAMPLES. [Chap. VIII. 

Ex. 7. At a station on the side of a pond, elevated 30 
feet above the water, the elevation of the summit of a cliff 
on the. opposite shore was found to be 37° 43' and the de- 
pression of the image 45° 26'. Required the elevation of 
the cliff. Ans. 221.8 ft. 

Ex. 8. To find the altitude of a tower on the brow of a 
hill, I measured, on slightly-inclined ground, a base-line 
AB 157 yards, A being on a level with the base of the 
hill. At A the angle of position of B and C was 87° 
45'; elevation of B, 2° 17'; of base of tower, 39° 43', and 
of top, 52° 13'. At B the depression of A was 2° 17'; the 
angle of position of A and C, 54° 23' ; elevation of base 
of tower, 33° 4', and of top, 45° 42'. Eequired the height 
of the hill and also of the tower. 

Ans. Height of hill, 172.5 ft.; of tower, 95.5 ft. 

Ex. 9. To determine the height of a tree C standing on 
the opposite shore Of a river, I measured a base-line AB of 
100 feet. At A the angle BAC was 90°, and the angle 
of depression of the image of the top of the tree was 39° 
48'. At B the angle of depression was 32°. Required the 
height, the instrument having been 10 feet above the water 
at each station. Ans. 84.47 feet. 

Ex. 10. Not being able to measure directly the three sides 
of a triangle, the corners of which were visible from each 
other, I took the angles as follow, — viz. : A = 57° 29', 
B = 72° 41', and C = 49° 50'. I also measured the dis- 
tances from the corners to a point within the triangle, and 
found them to be AD = 7.56 chains, BD = 9.43 chains, 
and CD = 8.42 chains. Required the lengths of the sides. 
Ans. AB = 12.63 chains, AC = 15.78 chains, and BC 
=s 13.94 chains. 

Ex. 11. The base of a triangle being 50 perches, and the 
area 5 acres, what are the other sides, their sum being S5 
perches ? Ans. 33.3785 P. and 51.6215 P. 

Ex. 12. It is required to lay out 7 acres in a triangular 
form, one side being 20 chains, and the others in the ratio 
of 2 to 3. 



MISCELLANEOUS EXAMPLES. 305 

Ans. The other sides are 9.86 and 14.79 chains, or 
39.58 and 59.37 chains. 

Ex. 13. The bearings of the dividing lines of two farms 
being as follow,— viz. : 1. K 83}° E. 2.37 chains ; 2. S. 47° 
E. 6.25 chains; 3. K 62f° E. 5.17 chains; 4. S. 56^° E. 
3.92 chains, and 5. "N. 14 J° E., — it is required to straighten 
the boundary, the new line to start from the beginning of 
the first side and fall on the last. The bearing of the new 
line is required, and also the distance on the last side. 

Ans. Bearing, S. 74° 40' E. to a point .25 chains back 
from the commencement of the last side. 

Ex. 14. One side of a tract running through a thick 
copse, I took a station S. 26J° E. 1.53 chains from the 
corner, and ran a "guess-line" bearing N". 60 J° E. 19.37 
chains, w T hen the other end bore K". 28 J° W. 3.27 chains. 
What is the course and distance of the line, and what must 
be the course and distance of an offset from a point 8.53 
chains on the random line, that it may strike a stone in the 
side 8.53 chains from the point of beginning? 

Ans. Side, K 55° 22' E. 19.42 chains ; 
Offset, K 28° 8' W. 2.29 chains. 

Ex. 15. Three observers, A, B, and C, whose distances 
asunder are AB = 1000 yards, BC = 1180 yards, and AC 
= 1690 yards, take the altitude of a balloon at the same 
instant, and find it to be as follow, — viz. : At A, 53° 43', 
at B, 46° 40', and at C, 52° 46'. Eequired the height of 
the balloon. Ans. 1461.4 yards or 2411 yards. 

Ex. 16. The bearings and distances of the sides of a tract 
of land are,— 1. 1ST. 61° 20' W. 22.55 chains; 2. K 10° W. 
16.05 chains ; 3. K 60° 45' E. 14.30 chains ; 4. S. 66° 40' E. 
17.03 chains ; 5. S. 86° E. 22.40 chains ; 6. S. 31° 40' E. 
19.10 chains, and 7. S. 76° 35' W. 39 chains,— to divide it 
into two equal parts by a line running due north. The 
position of the division line is desired. 

Ans. The division line runs from a point on the 7th 

side 3.77 chains from the end thereof. 

20 



306 MISCELLANEOUS EXAMPLES. [Chap. VIII. 

Ex. 17. Not being able to run a line directly, on account 
of a projecting cliff, I took the angles of deflection and the 
distances as follow, — viz. : 1. to the right, 67° 35' 10 chains ; 
2. to the left, 48° 43' 7.25 chains; 3. to the left, 11° 45' 
5.43 chains, and 4. to the left, 65° 17'. How far on the last 
course must I run before coming in line again? at what 
angle must I deflect to continue the former direction ? and 
what is the distance on the first line ? 

Ans. Distance on the last course, 14.42 chains ; on 
the first, 23.67 chains ; deflection, 58° 10' to the right. 

Ex. 18. To find the length of a tree leaning to the south, 
I measured due north from its base 70 yards, and found the 
elevation of the top to be 25° 10' ; then, measuring due 
east 60 yards, the elevation of the top was 20° 4'. What 
was the length and inclination of the tree ? 

Ans. Length, 35.1 yards ; inclination, 83° 11'. 

Ex. 19. The bearings and distances being as in Ex. 16, 
it is required to divide the tract into two equal parts by a 
line running from the first corner. The bearing of the 
division line is required. 

Ans. K 14° 59' E. 27.66 chains to a point on the fifth 
side 1.61 from beginning. 

Ex. 20. The boundaries of a quadrilateral are, — 1. "N. 35J° 
E. 23 chains ; 2. K 75|° E. 30.50 chains ; 3. S. 3J° E. 46.49 
chains, and 4. N". 66^° W. 49.64 chains, — to divide the tract 
into four equal parts by two straight lines, one of which 
shall be parallel to the third side. Required the distance 
of the parallel line from the first corner, the bearing of the 
other division line and its distance from the same corner, 
measured on the first side. 

Ans. Distance of parallel division, 32.50 chains ; bear- 
ing of the other, S. 88° 22' E. ; distance from the first 
corner, 5.99 chains. 



CHAPTER IX. 

MERIDIANS, LATITUDE, AND TIME. 



SECTION I. 
MERIDIANS. 

409. The meridian of a place is a true north and south 
Hue through that place ; or it may be denned to be a great 
circle of the earth passing through the pole and the place. 

410. As it is of great importance to the surveyor to 
be able to trace accurately a meridian line, the following 
methods are given. Any one of these is sufficiently accu- 
rate for his purposes. Those which require the employ- 
ment of the transit or the theodolite are to be preferred, 
if one of these instruments is at hand. When the obser- 
vations are performed with the proper care, and the instru- 
ments are to be depended on, the line may be run within a 
few seconds of its proper position. 

411. Although the methods to be explained in the follow- 
ing articles are in theory perfectly accurate, yet the results 
to which they lead cannot be relied on with the same cer- 
tainty when the observations are made with surveyors' 
instruments, as if the larger and more expensive instru- 
ments to be found in fixed observatories were employed. 
These instruments generally rest on permanent supports: 
their positions and adjustments may therefore be tested, and 
corrected when found defective, and thus their proper posi- 
tion be finally obtained with almost perfect accuracy. Not 

307 



308 



MERIDIANS, LATITUDE, AND TIME. 



[Chap. IX. 



so with the theodolite or the surveyors' transit. The ad- 
justments in their position must be made at the time, and 
renewed for every fresh observation. The results alone are 
to be corrected by subsequent observation, and not the 
'position of the instrument. Notwithstanding these diffi- 
culties, which must always prevent his attaining the pre- 
cision of the astronomer, yet, with ordinary care, the sur- 
veyor may run his lines with all the accuracy which is 
necessary for his operations. 

Problem 1. — -To run a meridian line. 

412. First Method. — By equal altitudes of the sun. 



Select a level surface, ex- Fi s- im- 

posed to the south, and erect 
an upright staff upon it. 
Around the foot of this staff 
A (Fig. 199) as a centre de- 
scribe a circle. Observe care- 
fully the point B at which the 
end of the shadow crosses this 
circle in the morning, and 
likewise the point C where it 
crosses in the evening. Bisect 
the angle BAC by the line NS, 
which will be a meridian. If 
a number of circles be de- 
scribed around A, several observations may be made on the 
same clay, and the mean of the whole taken. 

If the staff' is not vertical, let fall a plumb-line from the 
summit, and describe, the circles around the point in which 
this line cuts the surface. A piece of tin, with a small cir- 
cular hole through it for the sun's rays to pass through, is 
better than the top of the staff, the image being definite. 

Where much accuracy is not required, the above method 
is sufficient. It supposes the declination of the sun to re- 
main unchanged during the observation. This is not true 
except at the solstices,— 21st of June and 22d of December. 




Sec. L] MERIDIANS. 309 

Those days — or at least a time not very remote from them 
— should therefore be chosen for determining the meridian 
by this method. 

413. Second Method. — By a meridian observation of the 
North Star. 

Tlie Pole Star {Polaris, or a Ursce Minoris) is situated 
very nearly at the North Pole of the heavens. If it were 
exactly so, all that would be necessary to determine the 
direction of the meridian would be to sight to the star at 
anytime. The North Star, being, however, about 1J° from 
the pole, is only on the meridian twice in twenty-four 
hours. 

There is another star, {Alioth,) in the tail of the Great 
Bear, ( Ursai If aj oris,) which is on the meridian very nearly 
at the same time as the Pole Star. 

The constellation in which Alioth is situated is one of the 
most generally known. It is often called the Plough, the 
Dipper, the Wagon and Horses, or Charles's Wain. The 
two stars in the quadrangle farthest from the handle, or 
tail, are called the Pointers, from the fact that the line 
joining them will, when produced, pass nearly through the 
Pole Star. The star in the handle of the dipper, nearest 
the quadrangle, is Alioth. 

414. To determine the direction of the meridian. 

Suspend a long plumb-line from some fixed elevated 
point. If a window can be found properly situated, a staff 
may be projected from it to afford a support. The plum- 
met should be heavy, and be allowed to swing in a vessel 
of water, so as to lessen the effect of the currents in the 
air. At some distance to the south of the line set two posts, 
east and west from each other, making their tops level, and 
nail upon them a horizontal board. To another board 
screw a compass-sight. This may be moved steadily 
to the east or west upon the other board. Then, some 
time before Polaris is on the meridian, place the compass- 



310 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. 

sight so that by looking through it Alioth may be hidden 
by the plumb-line. As the star recedes from the line, 
move the sight, so as to keep the line and star in the same 
direction ; at last Polaris will also be covered by the line. 
The eye and plumb-line are then very nearly in the me- 
ridian. If the time is noted, and Polaris sighted to seven- 
teen minutes after the former observation, the meridian will 
be much more accurately determined. The compass-sight 
may now be firmly clamped till morning. In making the 
above described observation, it will generally be necessary 
for an assistant to illuminate the line if the night is dark. 

415. To determine the time Polaris is on the meridian. 

1. Take from the American Almanac, or other Ephemeris, 
the sun's right ascension, or sidereal time of mean noon, 
for the noon preceding the time for which the transit is 
wanted. The sidereal time is given in the American Al- 
manac for mean noon at Greenwich (England) for every day 
in the year, and may be calculated for any other meridian 
by interpolation, thus : — 

The difference between the sidereal times for two suc- 
cessive days being 3 minutes 56.555 seconds, say, As twenty- 
four hours is to the longitude expressed in time, so is 3 minutes 
56.555 seconds to the correction to be applied to the sidereal time 
at noon of the given day at Greenwich. This correction — 
added to the sidereal time taken from the almanac if the 
longitude be west, but subtracted if it be east — will give 
the sidereal time at mean noon at the given place. 

The above correction, having been once determined for 
the given place, will serve for all the calculations that may 
be wanted. 

Example. 

Ex. 1. Let it be required to find the sidereal time at 
mean noon, at Philadelphia, long. 5 h. m. 40 sec. W., on 
the 11th of August, 1855. 

The sidereal time at mean noon, Greenwich, August 11, 



Sec. L] MERIDIANS. 311 

is 9 hours, 17 minutes, 32.74 seconds, as taken from the 
American Almanac of that year. 

And, As 24 h. : 5 h. m. 40. s. : : 3m. 56M5 s. : 49.391. 

h. m. sec. 

Then, sidereal time at Greenwich, mean noon 9 17 32.74 
Correction for difference of long. 49.39 

Sidereal time at Philadelphia, mean noon 9 18 22.13 

2. Subtract the sidereal time above determined from the 
right ascension of the star, taken from the same almanac, 
increasing the latter by 24 hours, if necessary to make the 
subtraction possible. The remainder is the time of the 
transit expressed in sidereal hours. 

To convert these into solar hours. Say, As 24 hours is to 
the number of hours in the above time, so is 3 minutes 55.9 
seconds to the correction. This correction, subtracted from 
the sidereal time, will give the mean solar time of the upper 
transit. 

The time thus determined will be astronomical time. 
The astronomical day begins at noon, the hours being 
counted to twentvrfour. The first twelve hours, therefore, 
correspond with the hours in the afternoon of the same 
civil day ; but the last twelve agree with the hours of the 
morning of the next succeeding day. 

Thus, August 11, 8h. 15 m., astronomical time, corresponds 
with August 11, 8h. 15m. p.m., civil time; 
but August 11, 16 h. 15 m., astronomical time, agrees with 
August 12, 4 h. 15 m. a.m., civil time. 

If, therefore, the number of hours of a date expressed in 
astronomical time be greater than twelve, to convert it into 
civil time the days must be increased by one and the hours 
diminished by twelve. 

Required the time of the upper transit of Polaris, Sep- 
tember 11, 1855, for Philadelphia. 



312 MERIDIANS, LATITUDE, AND TIME. 

Sidereal time at mean noon, Greenwich, 

September 10 
Correction for Philadelphia 
Sidereal time, mean noon, atPhila. (A) 
Right ascension of Polaris, Sept. 11 (B) 
(B) - (A) 

Correction for 13 h. 50 m. 24 sec. 
Astronomical time, September 10 
agreeing with, civil time, Sept. 11 



[Chaj>. IX. 



h. m. sec. 

11 15 49.38 
49.39 



11 


16 38.77 


1 


7 2.71 


13 


50 23.94 




2 16.04 



13 48 7.90 
1 48 7.90 a.m. 



416. The times of the upper transit of Polaris for every 
tenth day of the year is given in the following table. 
The calculation is made for the meridian of Philadelphia, 
the year 1855. As a change of six hours, or 90° of longi- 
tude, will only make a change of one minute in the time 
of the transit, the table is sufficiently accurate for any place 
within the United States : — 

Time of Polaris crossing the meridian, upper transit. 



Months. 



January. . . . 
February . . 

March 

April 

May 

June 

July 

August 

September. 

October 

November. 
December.. 



1st. 



h. 

6 
4 
2 


10 
8 
6 
4 
2 


10 
8 



m. 

22 

20 

29 

27 

30 a 

28 

30 

29 

27 

30 

24 

26 



M. 



M. 



M. 



11th. 



5 43p 
3 40 
150 

1148 a 
9 50 
7 49 
5 51 
3 50 
148 

11 46 p 
9 44 
7 46 



M. 



M. 



M. 



21st. 



5 3p 
3 1 
111 
11 9a 
9 11 



7 
5 
3 
1 
11 
9 
7 



10 
12 
11 

9 

7p 

5 

7 



M. 



M. 



M. 



If the time of the passage of the star for any day not 
given in the table be desired, take out the time of passage 
for the day next preceding, and deduct four minutes for 



Sec. L] MERIDIANS. 313 

each clay that elapses between the date in the table and 
that for which the time of transit is required ; or, more ac- 
curately, thus : — 

Say, As the number of days between those given in the table is 
to the number between the preceding date and that for which the 
time of transit is desired, so is the difference between the times of { 
transit given in the table to the time to be subtracted from that 
corresponding to the earlier of the two days. 

Let the time of transit, August 27, be desired. 

Time. 

Aug. 21, 3 h. 11 m. 

Sept. 1, 2 27 , 

Difference 44 

As ' 11 d. : 6 d. : : 44 : 24 ; 

therefore 3 h. 11 m. — 24 m. = 2 h. 47 m. is the time re- 
quired. 

417. If the time of the lower transit be desired, it may 
be obtained from the table by changing a.m. into p.m. and 
diminishing the minutes by 2, or changing p.m. into a.m. and 
increasing the minutes by 2. 

418. The above table is calculated for the year 1855. It 
will, however, serve for the observation described in Art. 
414 for many years, the time of the meridian passage 
being determined in that method by the time of Polaris 
and Alioth being in the same vertical. "When the time is 
more accurately needed, as in Method 3 (Art. 419) for deter- 
mining the meridian, it will be necessary to correct the 
numbers in the table for the years that elapse between 1855 
and the current year. 

The Pole Star passes the meridian about 21 seconds — 
more accurately, 20.6 seconds — later every year than the 
preceding one, so that in 1860 the time will be 1 minute, 
43 seconds later than those given in the table ; in 1870, 5 
minutes ; in 1880, 8 minutes 35 seconds ; and, in 1890, 12 
minutes later. 



314 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. 

419. Third Method. — By a meridian passage observed with a 
transit or theodolite. 

Having accurately levelled the instrument, sight to Po- 
laris when on the meridian. Then, depressing the telescope, 
set up an object in the line of sight : a line drawn from the 
instrument to that object will be a meridian. 

In observing with the transit or theodolite at night, it is 
needful that the wires should be illuminated. This may be 
done by an assistant reflecting the rays of a lamp into the 
tube by a sheet of white paper. 

An error of 5 minutes in the time of the transit of Po- 
laris will make an error of about 1J' in the bearing of the 
star, so that if the observation is not made near the proper 
time, it must be corrected. 

This may be done thus : — Deduct the star's polar distance 
from the complement of the latitude. Then say, As sine 
of this difference is to the sine of the polar distance of the star, 
(1° 28' at present,) so is sine of the error in time (expressed in 
degrees) to the sine of the bearing of the star. East if the time 
be too early, but west if it be too late. 

The time is reduced to degrees by multiplying by 15 : 
thus, 5 minutes = 1° 15'. 

Example. 

Required the bearing of Polaris 5 minutes after the upper 
meridian passage, the latitude of the place being 40°. 

50° - 1° 28' = 48° 32' 

As sine of 48° 32' Ar. Co. 0.125320 

: sine of star's polar distance 1° 28' 8.408161 

: : sine of time, in degrees, 1° 15' 8.338753 

: sine of star's bearing 1' 37" W. 6.872234 

420. Fourth Method. — By an observation of Polaris at its 
greatest elongation. 

As a circumpolar star revolves round the pole, it gradu- 
ally recedes from the meridian towards the west until it 



Sec. I.] MERIDIANS. 315 

attains its most remote point: here it apparently remains 
stationary, or at least appears to move directly towards the 
horizon for a few minutes, and then gradually moves east- 
ward towards the meridian, which it crosses below the pole. 
Continuing its course, in about six hours it reaches its 
greatest eastern deviation, when it again becomes sta- 
tionary. When most remote from the meridian, it is said 
to have its greatest elongation. 

As the star is apparently stationary at the time of its 
greatest eastern or western elongation, this time is a very 
favorable one for observing it. A variation of a few 
minutes in the time will then make no appreciable error 
in the bearing of the line. 

421. The subjoined table contains the times of the great- 
est eastern or western elongations, according as the one or 
the other occurs at a time of day favorable for observa- 
tion. The times of greatest elongations are calculated 
thus : Take from one of the almanacs mentioned in Art. 415 
the polar distance of the star at the given time, and call it 
P. Call the latitude of the place L. Then find the semi- 
diurnal arc by the following formula: — 

H . cosine x = tan. P . tan. L. 

Reduce x to time by dividing by 15, calling the degrees 
hours, and correct for the sidereal acceleration : the result 
will be the semidiurnal arc expressed in time. Call it t 
Then, if T be the time of greatest elongation, and T r be 
the time of the upper meridian passage of the star, T = T' 
+ t or T' — t, according as the time of the western or 
eastern elongation is desired. 

The hour angle for Polaris at its greatest elongation, 
July 1, 1855, in lat. 40° N"., was 5 hours 54 minutes ; but, 
as the polar distance of the star is diminishing at the rate 
of 19. 23" per annum, the semidiurnal arc is slowly in- 
creasing. The change is so small, however, — being about 
one second per year, — that it may be entirely neglected. 
As the time of the meridian passage of the star is later by 
20.6 seconds each year than the preceding one, the times 



316 



MERIDIANS, LATITUDE, AND TIME. 



[Chap. IX. 



of greatest eastern and greatest western elongation will be 
similarly affected: in 1860 they will be 1 minute 43 seconds 
later than the times given in the table; in 1870, 5 minutes ; 
and, in 1880, 8 minutes 35 seconds later. 



422. Table of Times of Greatest Elongation of Polaris for 
1855. Latitude, 40° K 



Months. 




1st. 


nth. 


21st. 






h. m. 


h. m. 


h. m. 


January... 


"West 


16 A.M. 


11 37 p.m. 


10 57 p.m. 


February.. 


West 


10 14 p.m. 


9 35 " 


8 55 " 


March 


West 


8 23 " 


7 44 " 


7 4" 


April 


East 


6 33 a.m. 


5 54 a.m. 


5 15 A.M. 


May 


East 


4 35 " 


3 56 " 


3 17 « 




East 


2 34 " 


155 " 


1 15 " 


July 


East 


36 " 


11 53 p.m. 


11 14 P.M. 


August.... 


East 


1 10 31 p.m. 


9 51 " 


9 12 " 


September 


East 


8 29 " 


7 50 " 


7 11 " 


October ... 


West 


6 24 a.m. 


5 44 a.m. 


5 5 A.M. 


November 


West 


4 22 " 


3 42 " 


3 3" 


December 


West 


2 24 " 


145 " 


15" 



The above table is calculated for lat. 40°, for which lati- 
tude the hour angle is 5 h. 54 m. 6 sec. ; 
for latitude 50° the hour angle is 5 52 2, 
and for lat. 30° " " " 5 55 38 ; 
therefore, for lat. 50° the eastern elongation occurs two 
minutes later, and the western two minutes earlier, than 
those given in the table ; for lat. 30° the times of the 
eastern elongation must be diminished, and those of the 
western increased, by 1 minute 32 seconds. 

423. The observation for the meridian is made as directed 
Art. 414. Suspend the plumb-line, and, having placed the 
compass-sight on the table, as the star moves one way move 
the sight the other, so as to keep the star always hid by 
the line. At the time of greatest elongation the star will 
appear stationary behind the line. Clamp the board to 
which the compass-sight is attached. If the plumb-line is 
suspended from a point that is not liable to derangement, 



Sec. L] 



MERIDIANS. 



317 



the remainder of the work may be left till daylight ; other- 
wise, let an assistant take a short stake, with a candle 
attached to it, to a distance of 8 or 10 chains. He may 
then be placed exactly in line with the plumb. When the 
stake has been so adjusted, it should be driven firmly into 
the ground and its position again tested. 

Measure accurately the distance between the compass- 
sight and the stake. Call it D. Take the azimuth of the 
star from the following table and call it A. 

D . tan. A 
Calculate x = — , 

and set off the distance x to the east or west of the stake, 
according as the western or eastern elongation was' observed. 
The point thus determined will be on the meridian passing 
through the compass-sight. Permanent marks may then 
be fixed at any convenient points in this line. 

If a transit or theodolite is at hand, direct the telescope 
to the stake first set up. Turn it through an angle equal 
to the azimuth : it will then be in the meridian : or direct 
the telescope to the star when at its greatest elongation, 
and then turn the plate through an angle equal to the 
azimuth. 

424. The azimuth of a star is its bearing, and may be 
determined by the following formula, — A being the azi- 
muth, L the latitude of the place, and P the polar distance 
of the star: — 

a . A R . sin. P 

Sin. A = — . 

cos. L 



Azimuths of the Pole Star at its Greatest Elongation. 



Lat. 


1855. 


I860. 


1865. 


1870. 


o 


o / // 


O 1 II 


O 1 II 


o r ii 


30 


1 41 21 


1 39 32 


1 37 42 


1 35 49 


35 


1 47 11 


1 45 14 


1 43 16 


1 41 19 


40 


1 54 37 


1 52 32 


1 50 27 


1 48 20 


45 


2 4 11 


2 1 55 


1 59 35 


1 57 18 



318 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. 

The above are calculated from the mean place of the star 
as given in Loomis's "Practical Astronomy." 

425. Fifth Method. — By equal altitudes of a star. 

If a theodolite or a transit with a vertical arc is at hand, 
the meridian may be run very accurately by observing a 
star when at equal altitudes before and after passing the 
meridian. 

For this purpose select a star situated near the equator, 
and, having levelled the instrument with great care, take 
the altitude of the star about two or three hours before it 
passes the meridian, and notice carefully the horizontal 
reading. When the star is about as far to the west of the 
meridian, set the telescope to the same elevation, and fol- 
low the star by the horizontal motion until its altitude is 
the same as before, and again notice the reading. 

Then if the zero is not between the two observed read- 
ings, take half their sum, and turn the telescope until the 
vernier is at that number of degrees and minutes : the tele- 
scope will then be in the meridian. If the vernier has 
passed the zero, add 360 to the less reading before taking 
the sum. 

Thus, if the first reading were 150° 37' 30", and the 

431° 2 f 30" 

second 280° 25', the half sum = 215° 31' 15" 

' 2 

would be the reading for the meridian. 

Instead of taking the readings, a stake may be set up at 
any distance — say ten chains — in each observed course : then 
bisect the line joining the stakes, and run a line from the 
instrument to the point of bisection. 

The mean of a few observations taken in this manner 
will determine the meridian with considerable precision. 



Sec. II.] 



LATITUDE. 



319 



SECTION II. 

LATITUDE. 

The latitude of a place may be determined in various 
modes. 

426. First Method. — By a meridian altitude of the Pole Star. 

The altitude of the pole is equal to the latitude of the 
place. Take the altitude of Polaris when on the meridian, 
and from the result subtract the refraction taken from the 
following table. Increase or diminish the remainder by the 
polar distance of the star according as the lower or upper 
transit was observed : the result will be the latitude. 



427 



. Refraction to be taken from the apparent latitude. 



App. 
Alt. 

o 


Kef. 


App. 
Alt. 


Kef. 


App. 
Alt. 

o 


Kef. 


App. 

Alt. 

o 


Kef. 


App. 

Alt. 

o 


Kef. 


/ // 


o 


/ n 1 


/ 


II 


/ // 


20 


2 39 


30 


1 40 


40 


1 


9 


50 


49 


60 


34 


21 


2 30 


31 


1 37| 


41 


1 


7 


51 


47 


61 


32 


22 


2 23 


32 


1 33 


42 


1 


5 


52 


45 


62 


31 


23 


2 16 


33 


1 29| 


43 


1 


2 


53 


44 


63 


30 


24 


2 10 


34 


1 26 


44 


1 





54 


42 


64 


28 


25 


2 4[ 


35 


1 23 


45 





58 


55 


41 


65 


27 


26 


1 59 


36 


1 20 


46 





56 


5Q 


39 


m 


26 


27 


1 54 


37 


1 17 


47 





54 


57 


38 


67 


25 


28 


1 49 


38 


1 14 


48 





52 


58 


36 


68 


24 


29 


1 45| 


39 


1 12 ! 


49 





50 


59 


35 


69 


22 



428. Second Method. — Take the altitude of the star six 
hours before or after its meridian passage. The result, 
corrected for refraction, will be the latitude. 

429. Third Method. — By a meridian altitude of the sun. 



Take the meridian altitude of the upper or the lower 
limb of the sun, and correct for refraction. The result, 



320 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. 

increased or diminished by the semidiameter of the sun 
according as the lower or the upper limb was observed, will 
be the altitude of the sun's centre. (The apparent semi- 
diameter of the sun is given in the American Almanac for 
every day of the year.) 

To the altitude of the sun's centre, add his declination 
(taken from the same almanac) if south, but subtract it if 
north: the result subtracted from 90° will give the latitude. 

Instead of the sun, a bright star, the declination of which 
is small, may be observed. 

430. If the exact direction of the meridian is not known, 
the telescope must be fixed on the body some time before it 
is south. As the sun or star approaches the meridian its 
altitude increases, and it will therefore rise above the hori- 
zontal wire. Move the telescope in altitude and azimuth 
so as to follow the body until it ceases to leave the wire. 
The reading will then give the observed meridian altitude. 
The altitude alters very slowly for some minutes before and 
after its meridian passage, thus affording ample time to 
direct the telescope accurately towards the object. 

431. Fourth Method. — By an observation of a star in the 
prime vertical. 

Any great circle passing through the zenith is called a 
vertical circle. All such circles are perpendicular to the 
horizon. 

That vertical circle which is perpendicular to the meridian 
is called. the prime vertical: it cuts the horizon in the east 
and west points. 

Level the plates of the transit or theodolite carefully, and 
direct the telescope to the east or west, so that it may move 
in the prime vertical or nearly so. Then, having selected 
some bright star which passes the meridian a little south of 
the zenith, (the declination of such a star is rather less than 
the latitude of the place,) observe the time of its 'crossing 
the vertical wire of the telescope before passing the meridian, 
and again, when in the west, after its meridian passage. Let 



Sec. II.] LATITUDE. 321 

these times be called T and T'. Let tlie interval between 

T and T' be called x, which must be reduced to sidereal 

time by adding to the solar time 3 minutes 56.55 seconds 

for 24 hours, or 9.85 seconds per hour; also, let L be the 

latitude of the place, and D be the declination of the star. 

_ _ R. tan. D 
Then tan. L = 

COS. f X 

Thus, for example, the transit of a Lyrce over the prime 
vertical was observed July 1, 1855, at 10 h. 43 m. 4 sec, 
and again at 13 h. 3 m. 48 sec, mean solar time. Re- 
quired the latitude, — the apparent right ascension of the 
star (as given in the American Almanac) being 18 h. 32 m. 
4 sec, and the declination 38° 39' 0.4". 

Here the interval is 2 h. 20 m. 44 sec, solar time. 
Reduction 23 

2)2 21 7~ 





1 h. 10 m. 33.5 sec. = 17° 38' 2 


Cos. \x 


17° 38' 22" A. C. 0.020915 


tan. D 


38° 39' 0.4" 9.902940 


tan. L 


40° 0'4" 9.923855 



432. Half the sum of the observed times is the time of 
meridiem passage in mean solar time. If this is reduced to 
sidereal time and increased by the sidereal time of mean 
noon at the given place, the result should be equal to the 
right ascension of the star. 

In the example before us the times of observation are 

h. m. sec. 

10 43 4 

and 13 3 48 

Sum 2) 23 46 52 

Half sum 11 53 26 

Reduction for sidereal time 1 57 

(A) 11 55 23 

21 



322 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. 

Sidereal time, mean noon, at Greenwich 6 h. 35 m. 54 sec. 
Add for difference of meridians 







49 


6 


36 


33 


18 
18 


31 

32 


56 
4 



Add (A) 

Right ascension of star 

Error in position of the instrument 8" 

A slight error in the position of the instrument will make 
no appreciable error in the result. Hence, this method 
affords perhaps the best means of determining the latitude. 



SECTION III. 

TO FIND THE TIME OF DAY. 

433. First Method, — If a good meridian line has been 
run, the transit or theodolite may be placed in that line, 
and, being well levelled, the telescope, if adjusted by being 
directed to the meridian mark, will, when elevated, move in 
the meridian. 

Observe the time that the western limb of the sun comes 
to the vertical wire, and also when the eastern limb leaves 
it. The mean between these will be the time that the centre 
of the sun is on the meridian, or apparent noon. Increase or 
diminish the observed time of the passage of the centre 
by the equation of time according as the sun is too slow or 
too fast, and the result will be the time of mean noon as 
given by the watch. The difference between this and twelve 
hours will be the error of the watch. 

434. Second Method. — Calculate the time that a fixed star 
having but little declination will pass the meridian as 
directed for Polaris, Art. 415. Then the difference between 
the observed and the calculated time will be the error of 
the watch. 



Sec. III.] TO FIND THE TIME OF DAY. 323 

435. Third Method. — If the meridian line has not been 
determined, the time may be obtained by an altitude of the 
sun or of a star when out of the meridian. 

Take the altitude of the sun when three or four hours 
from the meridian, noting the time by the watch, and correct 
it for refraction and semidiameter. The altitude of the 
upper limb should be taken in the afternoon, and the 
lower in the morning, as the wire then crosses the face of 
the sun before the observation, and may be distinctly seen. 

Call the altitude of the sun A, the polar distance D, the 
latitude L, and the hour angle H. 

Then sin.* J H = °°«- * (A + L + D) sin. j(L ,+ D - A^ 

sin. D . cos. L 

or, if S = | (A + L + D), then S - A = \ (L + D - A), 

and sin.* J H = cos. 8 .sin. (8 -A^ 

sin. D . cos. L 



Rule. 

Call the corrected altitude A. From the Ephemeris take 
the sun's declination at the time of observation, (the watch- 
time will be sufficiently accurate) ; if north, subtract it from 
90°, but if south, add it to 90° : the result will be the sun's 
polar distance, which call D. Call the latitude of the place 
L. Let S = J (A + L + D). Add together Ar. Co. sin. D, 
Ar. Co. cos. L, cos. S, and sin. (S — A), divide the result 
by 2, and the quotient will be the sine of half the hour 
angle of the sun at the time of observation. If the obser- 
vation is made in the afternoon, the hour angle reduced to 
time is the apparent time ; but, if the observation is in the 
morning, the hour angle subtracted from 12 is the apparent 
time. To the apparent time apply the equation of time, 
and the result is the mean time of the observation. The 
difference between the calculated time and that shown by 
the watch is the error of the watch. 

Several observations may be made in the course of a few 
minutes, and the mean of the results taken. If the obser- 
vation is carefully made with a good transit or theodolite, 



324 MERIDIANS, LATITUDE, AND TIME. [Chap. IX. 

the time obtained by this method will not differ more than 
a small fraction of a minute from the true time. 

436. If a star is observed instead of the sun, the mode 
of calculation is the same. The hour angle will then be in 
sidereal hours, which must be converted into solar hours. 
The result, added to or subtracted from the time of the 
meridian passage of the star, according as the observation 
was made after or before the star had passed the meridian, 
will give the mean time of observation. 

437. If two altitudes of the sun or a star be taken, and 
the times noted by a watch, the true time and the latitude 
may both be found. But, as other and preferable methods 
have already been given for finding the latitude, it is un- 
necessary to give the rule here. 



CHAPTER X. 

VARIATION OF THE COMPASS.* 

438. It has been mentioned (Art. 268) that the magnetic 
and the geographical meridian do not generally coincide ; 
the difference between the directions of the two being 
called the variation of the compass. If this variation were 
constant, it would be of no practical importance to the sur- 
veyor. A line run by the compass at one time could be 
retraced on the same bearing at any other. The variation 
is, however, subject to continual changes, — some of them 
having a period of many years, perhaps several centuries, 
others being annual or diurnal, and some accidental or tem- 
porary. 



439. Secular Change. From the time of the earliest 
observations made in this country on the position of the 
magnetic needle till about the commencement of the pre- 
sent century, the north point was gradually moving to the 
west. Since then, the direction of its motion has been re- 
versed. This motion constitutes what is called the secular 
change. To give an idea of the extent of this deviation, 
the following table of observations, made at Paris, is pre- 
sented : — 



Tear. Variation. 

1541 7° East. 

1580 11 30' " 

1618 8 

1663 

1700 8 10 West. 

1780 19 55 " 

1805 22 5 " 

1814 22 34 " 



Year. Variation- 

1816 22° 25' West 

1823 22 23 

1827 22 20 

1828 22 5 

1829 22 12 

1835 22 3 

1853 20 17 



u 



it 



u 



* This subject, in its connections with Land Surveying, was first fully 
developed by Professor Gillespie, in his Treatise before referred to. 

325 



32G 



VARIATION OF THE COMPASS. 



[Chap. X. 



From this table, it appears that in 1580 the needle had 
attained its greatest eastern deviation. From that time to 
about the year 1814 it moved towards the west, the great- 
est deviation being 22° 34'. Since 1814 it has been moving 
to the east. 

From observations made at various places in Europe and 
America, it appears that similar changes have been going 
on throughout all these countries. 

440. The following table, mostly taken from the "Report 
of the Superintendent of the United States' Coast Survey" 
for 1855, gives the variation and secular change for some 
of the more important places in this country : — 













Change 


Locality. 


Lat. 


Lon. 


Date. 


Variation. 


in 1850. 


Montreal, C.W 


45° 30' 


73° 35' 


1850 


+ 9° 28' 


4-4' 




49° 40' 


79° 21' 


1850 


1° 36' 




Burlington, Vt 


44° 27' 


73° 10' 


1855 


9° 57'. 1 


4'.9 




43° 39' 


70° 16' 


1851 


11° 41/ 






44° 20' 


71° 2' 


1854 


9° 31' 


5'.2 


Providence, R.I — 


41° 50' 


71° 24' 


1855 


9° 31'.5 


6'.0 


New Haven, Conn. 


41° 18' 


72° 54' 


1845 


6° 17'.3 


4'. 8 




40° 43' 


74° 0' 


1845 


6° 25'.3 


5'.2 


Albany, N.Y 


42° 39' 


73° 44' 


1836 


6° 47' 


7'.2 


Philadelphia, Pa... 


39° 58' 


75° 10' 


1855 


4° 31'.7 


6'. 8 




40° 26' 


79° 58' 


1845 


33' 


3'.5 


Wilmington, Del... 


39° 45' 


75° 34' 


1846 


2° 30'.7 




Baltimore, Md 


39° 16' 


76° 34' 


1847 


2° 18'. 6 




Washington, D.C... 


38° 53' 


77° 1' 


1855 


2° 25' 


5'.0 


Petersburg, Va 


37° 14' 


77° 24' 


1852 


0° 26'. 5 






34° 


81° 2' 


1854 


— 3° 1'.7 






32° 5' 


81° 5' 


1852 


— 3° 40'. 3 






39° 6' 


84° 22' 


1845 


— 4° 4' 


4' 




39° 49' 


84° 47' 


1845 


— 4° 52' 


4' 


Detroit, Mich 


42° 24' 


82° 58' 


1840 


— 2° 0' 


V 


San Francisco, Cal. 


37° 48' 


122° 27' 


1852 


— 15° 27' 





The above are derived from the best data that could be 
procured ; but many of the observations are doubtless very 
imperfect. 



441. Line of no Variation. From a map published by 
Professor Loomis, it appears that in 1840 the lines of equal 
variation crossed the United States in a direction to the east 
of south, tending more to the east in the ISTew England 
States. At that date, the line of no variation passed a little 



VARIATION OF THE COMPASS. 327 

to the west of Pittsburg and to the east of Kaleigh, N.C., — 
all those portions of the country to the east of that line 
having western variation. From a similar map, published 
in the Report above referred to, it appears that the line 
of no variation had shifted to the west a few miles since 
that time. It also results from the calculations in the same 
report, that the rate of change in variation has now attained 
its maximum, and is beginning to diminish. 

442. As it is frequently of importance to know the 
former variation, the following information is added : — 



The variation in 

Burlington, Vt, in 1792 7° 38' W. 

Salem, Mass., 1781 7° 2' W. 

New Haven, Ct., 1761 5° 47' W. 

" 1819 4° 35' W. 

New York, 1686 8° 45' W. 

" " 1789 4° 20' W. 

Philadelphia, 1710 8° 30' W. 

1793 1° 30' ~W. 



1818; 7° 30' W. 
1805, 5° 57' W. 
1775, 5° 25' W. 

1750, 6° 22' W 
1824, 4° 40' W 
1750, 5° 45' \Y 
1837, 3° 52' \V 



443. Prom the table, (Art. 440,) the variation for any 
time not far remote from those given may readily be found. 
This will also apply for places not very far distant from the 
line of equal variation passing through that place. As, 
however, the rate of change varies, calculations based on 
such a table can only be considered correct when the 
interval of time is comparatively small. In all cases, when 
it can be done, the variation should be found by direct ob- 
servation by the methods explained in the next article. 

444. To determine the change in variation by old lines. 

As the rate of change varies, the above rule can only be 
considered as true when the interval of time has not been 
great. If a number of years have elapsed since the prior 
survey, and no observations can be found relating to the 
immediate neighborhood, the change of variation can be 



328 VARIATION OF THE COMPASS. [Chap. X. 

found, nearly, by comparison with other places where such 
observations have been made. 

"When any well-established marks can be found, the 
change may be determined by taking the bearings of these 
and comparing them with the records. The difference will 
give the change that has taken place between the dates of 
the two surveys. 

If the two marks are not on the same line, they may still 
be used for this purpose. Thus, according to an old deed, 
the bearings of three adjacent sides of a tract were as 
follows, — viz. : 1. Beginning at a marked locust, N". 60J° E. 
200 perches to a chestnut ; 2. E". 25J° E. 183 perches to a 
post; 3. K 45° E. 105.3 perches to a white-oak. The 
locust is gone, but the stump remains, and the white-oak is 
still standing. The intermediate corners are entirely lost. 

Setting the instrument over the stump, run "N. 60J° E. 200 
perches ; thence N". 25J° E. 183 perches ; and thence N". 45° 
E. 105.3 perches. 

If no change had taken place in the Fig. 200. 

variation, and both surveys had been 
accurately made, the last distance / y> D 

would have been terminated at the / / 

white-oak. Instead of this, however, / / 

the tree bears S. 54° 25' E. 2.93 perches. 
Fig. 200 is a draft of the above. 

From the bearings of AB, BC, and 
CD, calculate that of AD, which (Art. 
350) will be found to be N". 43° 59' E. 
470.38 perches. This, therefore, was 
the bearing and distance of AD at the 
time of the former survey. It is now the bearing and dis- 
tance of AD'. 

With the latitude and. departure of AD' and that of DD', 
calculate the present bearing and distance of AD (Art. 350.) 
It will be found to be 1ST. 47° 54' E. 476.25 perches. The 
change of variation has therefore been 3° 55 f W. There is 
likewise a variation of 5.87 perches in the measurement, 
from which it is inferred that the chain used in the former 
survey was 101.25 links in length, or 1J links too long. 



VARIATION OF THE COMPASS. 329 

In order, therefore, correctly to trace the lines of the tract, 
the vernier of the compass must be set 3° 55' W., and all 
the distances be increased 1J- links per chain, or 1J perches 
per hundred. The magnetic bearings and the distances of 
the three sides are now, — 1. 1ST. 64° 25' E. 202.5 perches; 
2. K 29° 10' E. 185.3 perches; 3. K 48° 55' E. 106.6 
perches. 

445. Diurnal Change. If the position of the needle be 
accurately noted at sunrise on a clear summer day, and the 
observation be repeated at intervals, it will be found that 
the north pole will gradually be deflected to the west, attain- 
ing its maximum deviation about 2 or 3 o'clock. During 
the afternoon it will gradually return towards its former 
position, which it will regain about 8 or 9 o'clock in the 
evening. This deviation from the normal position is 
known as the diurnal change. It amounts sometimes to as 
much as a quarter of a degree, being greater in a clear day 
than when the sky is overcast, and not being perceptible 
if the day is entirely cloudy. It is likewise greater in 
summer than in winter. 

In consequence of this diurnal change, it is evident that 
a line run in the morning cannot be retraced with the 
same bearings at noon. The surveyor should therefore 
record not merely the date at which a survey is made, but 
also the time of day at which any important line was run, 
and also the state of the weather, whether clear or other- 
wise. 

446. Irregular Changes. Besides the seculat and 
diurnal changes, the needle is subject to disturbance from 
the passage of thunder storms, or from the occurrence of 
aurora boreali. It is likewise sometimes violently agitated 
when no apparent cause exists. Such disturbances pro- 
bably result from the occurrence of a distant magnetic 
storm, which would otherwise be unperceived, or from the 
passage of electric currents through the atmosphere. 

447. From the preceding articles it will be apparent that 



330 VARIATION OF THE COMPASS. [Chap.X. 

the needle, though an invaluable instrument for many pur- 
poses, is little to be depended on where precision is re- 
quired. It would be very desirable that prominent marks, 
the bearings of which were fully known, were established 
over the country, and that all important lines should be 
determined, by triangulation, from these. The true bear- 
ings should always be recorded. There would then be no 
difficulty in retracing old lines. In the State of Pennsyl- 
vania, and perhaps in some others, this is now required by 
law, though it is very doubtful whether the law is yet car- 
ried out in a way to be of much practical benefit, owing to 
the want of scientific knowledge on the part of much 
the larger number of those who undertake the business of 
surveying. 

Until there is a more general diffusion of theoretical as 
well as practical science among those whose business it is 
to settle the boundaries of estates, cases will continually 
occur in which confusing lines will be found to exist. This 
could never occur if all the bearings were made to the true 
meridian, the surveyor being careful to determine the local 
attraction and to allow for it in making his record. In no 
instance should a station be left before the back-sight has 
been taken, since, even in those regions where but little 
such influence exists, it will sometimes be found at par- 
ticular points. It sometimes likewise extends, without any 
change, over a considerable space, and thus may deflect 
the needle similarly at a number of stations. An instance 
of this kind was related to the author, a short time since, 
by a surveyor of great practical experience. 

A line was in dispute. One of the parties called in a 
surveyor, whom we shall call A., who ran the line, coming 
out at a stone. The other party, not being satisfied, called 
upon B., who traced a line agreeing exactly with the one run 
by A. until he came to a certain point: he then deviated 
from the former line some 4° to the west. He likewise ar- 
rived at a stone. Both parties were now dissatisfied. The first 
called on A. again, who retraced his line, following exactly 
his former course. B. was again employed. His course de- 
viated at the same point as before from A.'s. It was then 



VARIATION OF THE COMPASS. 331 

concluded to have them together. B., being the older 
hand, went ahead. When they arrived at the point at 
which their lines separated, B. called on A. to look through 
the sights, saying, "Is not this right, Mr. A. ?" " It looks 
very well," he replied: "but look back, Mr. B." On 
doing so, he found he was really running 4° to the west of 
his former course. The attraction was first manifest at 
that point, and continued, without change, at all the sub- 
sequent stations along the line he had traversed. 



APPENDIX. 



The following demonstration of the rule for finding the area of a triangle 
when three sides are given is more concise than that given in Art. 251. As 
the former, however, develops some important properties respecting the centre 
of the inscribed circle, it was thought best to retain it : — 



Let ABC (Fig. 201) be the triangle, 
the construction being the same as in 
Fig. 50, p. 75. 

Then, as was proved in the demon- 
stration of the Rule in Art. 143, 

AK = £ (AB -f BC + AC) = £ s. 
AI = J s — BC. 



Fig. 201. 




We have also 

KD = Bl = £ s — AC, and KB = J s — AB. 

Now, from similar triangles, ADE and AFB, we have 

AE : ED : : AF : FB. 
But AF : ED : : AF : ED ; 

whence (23.6) AE . AF : ED 2 : : AF 2 : ED . FB. 

But AE . AF = AK . AI (Cor. 36.3), 

and ED . FB = HB . FB = IB . BK (35.3) ; 



AI . AK : ED a : : AF 3 : IB . BK, 



and 



y/ AI . AK . IB. BK = ED . AF = ED . (AE -f EF) 

= ADC + BDC = ABC. 



332 



MATHEMATICAL TABLES. 



MATHEMATICAL TABLES. 



PAGE 

I. Table of Latitudes and Departures * 3 

II. Table of Logarithms of Numbers »........<,... 17 

III. Table of Logarithmic Sines and Tangents Bo 

IY. Table of Natural Sines and Cosines 87 

V. Table of Chords 97 



TRAYERSE TABLE; 



OR, 



DIFFERENCE OF LATITUDE 



AND 



DEPARTURE. 





_ • — — 

LATITUDES AZffD DEFAB.TUB.ES. 




D. 

| 1 


i Deg. j 


1 1 

i Deg. 


I Deg. 


1 Deg. 


D. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




I.OOOO 


.0044 ! 


I.0000 


.0087 


•9999 


1 
.0131 


.9998 


.0175 


1 




1 2 


2.0000 


.0087 


I.9999 


.0175 


1.9998 


.0262 


I.9997 


.0349 


2 




! 3 


3.0000 


.0131 


2.9999 


.0262 


2.9997 


•0393 


2.9995 


.0524 


3 




4 


4.0000 


.0175 


3.9998 


.0349 


3-9997 


.0524 


3-9994 


.0698 


4 




5 


5.0000 


.0218 


4.9998 


.0436 


4.9996 


.0654 


4.9992 


.0873 


5 




6 


5.9999 


.0262 


5.9998 


.0524 


5-9995 


.0785 


5-9991 


.1047 


6 




! 7 


6.9999 


.0305 


6.9997 


.0611 


6.9994 


.0916 


6.9989 


.1222 


7 




8 


7.9999 


•°349 


7.9997 


.0698 


7-9993 


.1047 


7.9988 


.1396 


8 




9 


8.9999 


•°393 


8.9997 


.0785 


8.9992 


.1178 


8.9986 


.1571 


9 




10 

1 

1 


9.9999 


.0436 


9.9996 


.0873 


9.9991 


.1309 


9.9985 


•1745 


10 




89f Deg. 


89 J Deg. 


89 £ Deg. 


89 Deg. 




U Deg. 


1* Deg. 


If Deg. 


2 Deg. 




.9998 


.0218 


•9997 


.0262 


•9995 


.0305 


•99'94 


.0349 


1 




2 


1.9995 


.0436 


1.9993 


.0524 


1.9991 


.0611 


1.9988 


.0698 


2 




3 


2.9993 


.0654 


2.9990 


.0785 


2.9986 


.0916 


2.9982 


.IO47 


3 




4 


3.9990 


.0873 


3.9986 


.1047 


3.9981 


.1222 


3-9976 


.1396 


4 




5 


4.9988 


.1091 


4.9983 


.1309 


4-9977 


.1527 


4.9970 


•1745 


5 




6 


5.9986 


.1309 


5-9979 


.1571 


5-9972 


.1832 


5.9963 


.2094 


6 




7 


6.9983 


.1527 


6.9976 


.1832 


6.9967 


.2138 


6.9957 


.2443 


7 




8 


7.9981 


•1745 


7.9973 


.2094 


7.9963 


.2443 


7-995 1 


.2792 


8 




9 


8.9979 


.1963 


8.9969 


.2356 


8.9958 


.2748 


8.9945 


•3 J 4i 


9 




10 
1 


9.9976 


.2181 


9.9966 


.2618 


9-9953 


•3°54 


9.9939 


•349° 


10 
1 




88f Deg. 


88| Deg. 


88i Deg. 


88 Deg. 




2i Deg. 


2i Deg. 


2f Deg. 


3 Deg. 




.9992 


•°393 


.9990 


.0436 


.9988 


.0480 


.9986 


.0523 




2 


1.9985 


.0785 


1. 9981 


.0872 


1.9977 


.0960 


1.9973 


.1047 


2 




3 


2.9977 


.1178 


2.9971 


.1308 


2.9965 


.1439 


2.9959 


•157° 


3 




4 


3.9969 


.1570 


3.9962 


•1745 


3-9954 


.1919 


3-9945 


.2093 


4 




5 


4.9961 


.1963 


4.9952 


.2181 


4.9942 


•2399 


4-993 1 


.2617 


5 




6 


5-9954 


.2356 


5-9943 


.2617 


5-993 1 


.2879 


5.9918 


.3140 


6 




7 


6.9946 


.2748 


6.9933 


•3°53 


6.9919 


•335 8 


6.9904 


.3664 


7 




8 


7.9938 


.3140 


7.9924 


•349° 


7.9908 


.3838 


7.9890 


.4187 


8 




9 


8.9931 


•3533 


8.9914 


.3926 


8.9896 


.4318 


8.9877 


.4710 


9 




10 
1 


9-99*3 


.3926 


9.9905 


.4362 


9.9885 


.4798 


9.9863 


.5234 


10 
1 




871 Deg. 


87i Deg. 


87i Deg. 


87 Deg. 




3 i Deg. 


3* Deg. 


3| Deg. 


4 Deg. 




.9984 


.0567 


.9981 


.0610 


•9979 


.0654 


.9976 


.0698 




2 


1.9968 


•"34 


1.9963 


.1221 


1-9957 


.1308 


1.9951 


•1395 


2 




3 


2.9952 


.1701 


2.9944 


.1831 


2.9936 


.1962 


2.9927 


.2093 


3 




4 


3.9936 


.2268 


3.9925 


.2442 


3-99*4 


.2616 


3.9903 


.2790 


4 




5 


4.9920 


.2835 


4.9907 


.3052 


4.9893 


.3270 


4.9878 


.3488 


5 




6 


5.9904 


.3402 


5.9888 


.3663 


5.9872 


•39 2 4 


5«9 8 54 


.4185 


6 




7 


6.9887 


.3968 


6.9869 


.4273 


6.9850 


.4578 


6.9829 


.4883 


7 




8 


7.9871 


•4535 


7.9851 


.4884 


7.9829 


.5232 


7.9805 


•558i 


8 




9 


8.9855 


.5102 


8.9832 


•5494 


8.9807 


.5886 


8.9781 


.6278 


9 




10 


9.9839 


.5669 


9.9813 


.6105 


9.9786 


.6540 


9.9756 


.6976 


10 

D. 




D. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




86f Deg. 


86£ Deg. 


861 Deg. 


86 Deg. 





22 



LATITUDES AND DEPARTURES. 




D. 

1 


41 Deg. 


4£ Deg. 


4f Deg. 


5 Deg. 


I 
D. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




•9973 


.0741 


.9969 


•0785 


.9966 


.0828 


.9962 


.0872 


1 




! % 


1.9945 


.1482 


I.9938 


.1569 


I.9931 


.1656 


I.9924 


•1743 


2 




1 3 


2.9918 


.2223 


2.9908 


.2354 


2.9897 


.2484 


2.9886 


.2615 


3 




1 4 


3.9890 


.2964 


3-9877 


.3138 


3.9863 


•33 12 


3.9848 


.3486 


4 




! 5 


4.9863 


•3705 


4.9846 


•392-3 


4.9828 


.4140 


4.9810 


•4358 


5 




1 6 


5-9^35 


•4447 


5-98I5 


.4708 


5-9794 


.4968 


5.9772 


.5229 


6 




! 7 


6.9808 


.5188 


6.9784 


•5492 


6.9760 


•5797 


6.9734 


.6101 


7 




8 


7.9780 


.5929 


7-9753 


.6277 


7.9725 


.6625 


7.9696 


.6972 


8 




9 


8-9753 


.6670 


8.9723 


.7061 


8.9691 


•7453 


8.9658 


•7844 


9 




10 
1 


9.9725 


.7411 


9.9692 


.7846 


9.9657 


.8281 


9.9619 


.8716 


10 
1 




85f Deg. 


85J Deg. 


851 Deg. 


85 Deg. 




51 Deg. 


5| Deg. 


5f Deg. 


6 Deg. 




.9958 


.0915 


•9954 


.0958 


.9950 


.1002 


•9945 


.1045 




2 


1. 9916 


.1830 


1.9908 


.1917 


1.9899 


.2004 


1.9890 


.2091 


2 




3 


2.9874 


.2745 


2.9862 


.2875 


2.9849 


.3006 


2.9836 


.3136 


3 




4 


3-9 8 3 2 


.3660 


3.9816 


•3834 


3-9799 


.4008 


3.9781 


.4181 


4 




5 


4.9790 


•4575 


4.9770 


.4792 


4.9748 


.5009 


4.9726 


.5226 


5 




6 


5-9748 


.5490 


5.9724 


•5751 


5.9698 


.6011 


5.9671 


.6272 


6 




7 


6.9706 


.6405 


6.9678 


.6709 


6.9648 


.7013 


6.9617 


.7317 


7 




8 


7.9664 


.7320 


7.9632 


.7668 


7-9597 


.8015 


7.9562 


.8362 


8 




9 


8.9622 


.8235 


8.9586 


.8626 


8.9547 


.9017 


8.9507 


.9408 


9 




10 
1 


9.9580 


.9150 


9.9540 


•9585 


9.9497 


1. 0019 


9.9452 


1.0453 


10 




84f Deg. 


84£ Deg. 


841 Deg. 


84 Deg. 


1 




61 Deg. 


6* Deg. 


6f Deg. 


7 Deg. 




-9941 


.1089 


.9936 


.1132 


.9931 


.1175 


.9925 


.1219 




2 


1.9881 


.2177 


1.9871 


.2264 


1. 9861 


.2351 


1.9851 


•2437 


2 




3 


2.9822 


.3266 


2.9807 


•339 6 


2.9792 


.3526 


2.9776 


•3656 


3 




4 


3.9762 


•4355 


3-9743 


.4528 


3-97^3 


.4701 


3.9702 


•4875 


4 




5 


4.9703 


•5443 


4.9679 


.5660 


4-9 6 53 


•5877 


4.9627 


.6093 


5 




6 


5-9 6 43 


.6532 


5.9614 


.6792 


5.9584 


.7052 


5-9553 


.7312 


6 




7 


6.9584 


.7621 


6.9550 


.7924 


6.9515 


.8228 


6.9478 


•853i 


7 




8 


7.9524 


.8709 


7.9486 


.9056 


7-9445 


•9403 


7.9404 


.9750 


8 




9 


8.9465 


.9798 


8.9421 


1.0188 


8.9376 


1.0578 


8.9329 


1.0968 


9 




10 
1 


9.9406 


1.0887 


9-9357 


1. 1320 


9.9307 


1. 1754 


9-92-55 


1. 2187 


10 
1 




83| Deg. 


83 £ Deg. 


831 Deg. 


83 Deg. 




71 Deg. 


7£ Deg. 


7f Deg. 


8 Deg. 




.9920 


.1262 


.9914 


•i3°5 


.9909 


.1349 


.9903 


.1392 




2 


1.9840 


.2524 


1.9829 


.2611 


1.9817 


.2697 


1.9805 


.2783 


2 




3 


2.9760 


.3786 


2.9743 


.3916 


2.9726 


.4046 


2.9708 


•4175 


3 




4 


3.9680 


.5048 


3-9 6 58 


- .5221 


3-9635 


•5394 


3.9611 


•5567 


4 




5 


4.9600 


.6310 


4.9572 


.6526 


4-9543 


•6743 


4-95I3 


.6959 


5 




6 


5.9520 


.7572 


5-9487 


.7832 


5-945 2 


.8091 


5.9416 


.8350 


6 




7 


6.9440 


.8834 


6.9401 


•9 J 37 


6.9361 


.9440 


6.9319 


.9742 


7 




8 


7.9360 


1.0096 


7.9316 


1.0442 


7.9269 


1.0788 


7.9221 


1.1134 


8 




9 


8.9280 


I-I358 


8.9230 


1. 1747 


8.9178 


1.2137 


8.9124 


1.2526 


9 




10 

D. 


9.9200 


1.2620 


9.9144 


1.3053 


9.9087 


1.3485 


9.9027 


1. 3917 
Lat. 


10 

D. 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




82| Deg. 


82 i Deg. 


821 Deg. 


82 Deg. 





LATITUDES AND DEPARTURES. 


D. 
1 


81 Deg. 


81 Deg. 


8f Deg. 


9 Deg. 


D. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


.9897 


•1435 


.9890 


.1478 


.9884 


.1521 


.9877 


.1564 


1 


2 


1-9793 


.2870 


I.9780 


.2956 


I.9767 


.3042 


1-9754 


.3129 


At 


3 


2.9690 


.4305 


2.9670 


•4434 


2.9651 


.4564 


2.9631 


.4693 


3 


4 


3.9586 


.5740 


3-95 6 l 


.5912 


3-9534 


.6085 


3.9508 


.6257 


4 


5 


4.9483 


•7175 


4.9451 


.7390 


4.9418 


.7606 


4.9384 


.7822 


5 


6 


5-9379 


.8610 


5-9341 


.8869 


5.9302 


.9127 


5.9261 


.9386 


6 


7 


6.9276 


I.0044 


6.9231 


1.0347 


6.9185 


1.0649 


6.9138 


I.0950 


< 


8 


7.9172 


"479 


7.9121 


1. 1825 


7.9069 


1. 2170 


7.9015 


I.2515 


8 


9 


8.9069 


1. 2914 


8.9011 


1.3303 


8.8953 


1. 3691 


8.8892 


I.4079 


9 


10 

i 
i 

1 

1 


9.8965 


1-4349 


9.8902 


1.4781 


9.8836 


1. 5212 


9.8769 


I.5643 


10 
1 


81f Deg. 


811 Deg. 


811 Deg. 


81 Deg. 


91 Deg. 


91 Deg. 


9f Deg. 


10 Deg. 


.9870 


.1607 


.9863 


.1650 


.9856 


.1693 


.9848 


.1736 


2 


1.9740 


•3 2I 5 


1.9726 


.3301 


1.9711 


.3387 


1.9696 


•3473 


2 


3 


2.9610 


.4822 


2.9589 


.4951 


2.9567 


.5080 


2.9544 


.5209 


3 


4 


3.9480 


.6430 


3-9451 


.6602 


3.9422 


.6774 


3-9392- 


.6946 


4 


5 


4.9350 


.8037 


4-93I4 


.8252 


4.9278 


.8467 


4.9240 


.8682 


5 


6 


5.9220 


.9645 


5-9*77 


.9903 


5-9*33 


I.0161 


5.9088 


1. 0419 


8 


7 


6.9090 


1. 1252 


6.9040 


«553 


6.8989 


I-I854 


6.8937 


1.2155 


7 


8 


7.8960 


1.2859 


7.8903 


1.3204 


7.8844 


I-3548 


7-8785 


1.3892 


8 


9 


8.8830 


1.4467 


8.8766 


1.4854 


8.8700 


I.5241 


8.8633 


1.5628 


9 


10 
1 


9.8700 


1.6074 


9.8629 


1.6505 


9.8556 


I.6935 


9.8481 


I-7365 


10 


80f Deg. 


801 Deg. 


801 Deg. 


80 Deg. 


1 


101 Deg. 


101 Deg. 


lOf Deg. 


11 Deg. 


.9840 


.1779 


•9 8 33 


.1822 


.9825 


.1865 


.9816 


.1908 


2 


1. 9681 


•3559 


1.9665 


•3 6 45 


1.9649 


•373° 


i-9 6 33 


.3816 


2 


3 


2.9521 


•533* 


2.9498 


.5467 


2.9474 


.5596 


2.9449 


•57M 


3 


4 


3.9362 


.7118 


3-933° 


.7289 


3.9298 


.7461 


3.9265 


.7632 


4 


5 


4.9202 


.8897 


4.9163 


.9112 


4.9123 


.9326 


4.9081 


.9540 


5 


6 


5.9042 


1.0677 


5- 8 995 


1.0934 


5-8947 


1.1191 


5.8898 


1. 1449 


6 


7 


6.8883 


1.2456 


6.8828 


1.2756 


6.8772 


i-3°57 


6.8714 


1-3357 


7 


8 


7.8723 


1.4235 


7.8660 


1-4579 


7.8596 


1.4922 


7.8530 


1.5265 


8 


9 


8.8564 


1. 6015 


8.8493 


1.6401 


8.8421 


1.6787 


8.8346 


1. 7173 


9 


10 
1 


9.8404 


1.7794 


9.8325 


1.8224 


9.8245 


1.8652 


9.8163 


1. 9081 


10 


791 Deg. 


791 Deg. 


791 Deg. 


79 Deg. 




Ill Deg. 


Ill Deg. 


Ill Deg. 


12 Deg. 


.9808 


.1951 


•9799 


.1994 


.9790 


.2036 


•9781 


.2079 


1 


2 


1. 9616 


.3902 


1.9598 


•39 8 7 


1.9581 


•4°73 


1-9563 


.4158 


2 


3 


2.9424 


•5853 


2.9398 


.5981 


2.9371 


.6109 


2.9344 


.6237 


3 


4 


3-923 1 


.7804 


3.9197 


•7975 


3.9162 


.8146 


3.9126 


.8316 


4 


5 


4.9039 


•9755 


4.8996 


.9968 


4.8952 


1. 0182 


4.8907 


1.0396 


5 


6 


5.8847 


1.1705 


5-8795 


1. 1962 


5-8743 


1. 2219 


5.8689 


1.2475 


6 


7 


6.8655 


1.3656 


6.8595 


1.3956 


6.8533 


1.4255 


, 6.8470 


1-4554 


7 


8 


7.8463 


1.5607 


7.8394 


1.5949 


7.8324 


1. 6291 


7.8252 


1.6633 


8 


9 


8.8271 


1.7558 


8.8193 


1-7943 


8.8114 


1.8328 


8.8033 


1. 8712 


9 


10 


9.8079 


1.9509 


9.7992 


1.9937 


9.7905 


2.0364 


9.7815 


2.0791 
Lat. 


10 

- 

D. 


D. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


781 Deg. 


781 Deg. 


781 Deg. 


78 Deg. 



LATITUDES AND DEPARTURES. 




1 
1 


121 Deg. 


12* Deg. 


12| Deg. 


13 Deg. 


D. 
1 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




•9772 


1 
.2122 


•97 6 3 


.2164 


•9753 


.2207 


•9744 


.2250 




; « 


1-9545 


.4244 


1.9526 


•43 2 91 


1.9507 


•44 1 4 


1.9487 


•4499 


2 




3 


2.9317 


.6365 


2.9289 


• 6 493 


2.9260 


.6621 


2.9231 


.6749 


3 




4 


3.9089 


.8487 


3.9052 


.8658 


3.9014 


.8828 


3- 8 975 


.8998 


4 




5 


4.8862 


I.0609 


4.8815 


1.0822 


4.8767 


1. 1035 


4.8719 


1. 1 248 


5 




6 


5.8634 


I.273I 


5-8578 


1.2986 1 


5.8521 


1.3242 


5.8462 


r -3497 


6 




7 


6.8406 


I.4852 


6.8341 


1.5151 


6.8274 


1-5449! 


6.8206 


J -5747 


7 




8 


7.8178 


I.6974 


7.8104 


i-73 x 5 


7.8027 


1.7656 1 


7.7950 


1.7996 


8 




9 


8.7951 


1.9096 j 


8.7867 


1.9480 


8.7781 


1.9863 


8.7693 


2.0246 


9 




10 
1 


9.7723 


2.1218 


9.7630 


2.1644 


9-7534 


2.2070 


9-7437 


2.2495 


10 
1 




77f Deg. 


77* Deg. 


77i Deg. 


77 Deg. 




13i Deg. 


13* Deg. 


13f Deg. 


14 Deg. 




•9734 


.2292 


.9724 


• 2 334 


•9713 


•2377 


•9703 


.2419 




2 


1.9468 


.4584 


1.9447 


.4669 


1.9427 


•4754 


1.9406 


.4838 


2 




3 


2.9201 


.6876 


2.9171 


.7003 


2.9140 


•7I3 1 


2.9109 


.7258. 


3 




4 


3- 8 935 


.9168 


3.8895 


•933 8 


3.8854 


.9507 


3.8812 


.9677 


4 




5 


4.8669 


1. 1460 


4.8618 


1. 1672 


4.8567 


1. 1884 


4- 8 5*5 


1.2096 


5 




6 


5.8403 


I-375 2 


5.8342 


1.4007 


5.8281 


1. 4261 


5.8218 


1.4515 


6 




7 


6.8137 


1.6044 


6.8066 


1. 6341 


6.7994 


1.6638 


6.7921 


I-6935 


7 




8 


7.7870 


1.8336 


7.7790 


1.8676 


7.7707 


1.9015 


7.7624 


1-9354 


8 




9 


8.7604 


2.0628 


8 -75 J 3 


2.1010 


8.7421 


2.1392 


8.7327 


2.1773 


9 




< 10 
1 


9-733 8 


2.2920 


9.7237 


2 -3345 


9-7*34 


2.3769 


9.7030 


2.4192 


10 




76| Deg. 


76* Deg. 


76i Deg. 


76 Deg. 


1 




14i Deg. 


141 Deg. 


14| Deg. 


15 Deg. 




.9692 


.2462 


.9681 


.2504 


.9670 


.2546 


.9659 


.2588 




2 


1.9385 


•4923 


1.9363 


.5008 


1-9341 


.5092 


1. 9319 


•5176 


2 




3 


2.9077 


.7385 


2.9044 


.7511 


2.9011 


.7638 


2.8978 


•7765 


3 




4 


3.8769 


.9846 


3.8726 


1.0015 


3.8682 


1. 0184 


3.8637 


I -°353 


4 




5 


4.8462 


1.2308 


4.8407 


1.2519 


4.8352 


1.2730 


4.8296 


1.2941 


5 




6 


5- 8l 54 


1.4769 


5.8089 


1.5023 


5.8023 


1.5276 


5-7956 


1.5529 


6 




7 


6.7846 


1. 7231 


6.7770 


1.7527 


6.7693 


1.7822 


6.7615 


1.8117 


7 




8 


7.7538 


1.9692 


7.7452 


2.0030 


7.7364 


2.0368 


7.7274 


2.0706 


8 




» 


8.7231 


2.2154 


8 -7!33 


2.2534 


8.7034 


2.2914 


8.6933 


2.3294 


9 






9.6923 


2.4615 


9.6815 


2.5038 


9.6705 


2.5460 


9-6593 


2.5882 


10 

1 




75f Deg. 


75* Deg. 


75i Deg. 


75 Deg. 




15i Deg. 


15* Deg. 


15f Deg. 


16 Deg. 




.9648 


.2630 


.9636 


.2672 


.9625 


.2714 


.9613 


.2756 




! 2 


1.9296 


.5261 


1.9273 


•5345 


1.9249 


.5429 


1.9225 


•55*3 


2 




3 


2.8944 


.7891 


2.8909 


.8017 


2.8874 


.8143 


2.8838 


.8269 


3 




1 4 


3- 8 59* 


1. 0521 


3- 8 545 


1.0690 


3.8498 


1.0858 


3- 8 45° 


1. 1025 


4 




1 5 


4.8239 


1. 3152 


4.8182 


1.3362 


4.8123 


I-357 2 


4.8063 


1.3782 


5 




6 


5.7887 


1.5782 


5.7818 


1.6034 


5-7747 


1.6286 


5.7676 


1.6538 


6 




1 7 


6-7535 


1. 8412 


6.7454 


1.8707 


6.7372 


1. 9001 


6.7288 


1.9295 


7 




8 


7-71*3 


2.1042 


7.7090 


2.1379 


7.6996 


2.1715 


7.6901 


2.2051 


8 




9 


8.6831 


2.3673 


8.6727 


2.4051 


8.6621 


2.4430 


8.6514 


2.4807 


9 




10 


9.6479 


2.6303 


9.6363 


2.6724 


9.6246 


2.7144 


9.6126 


2.7564 


10 

D. 




D. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




74| Deg. 


74* Deg. 


74£ Deg. 


74 Deg. 





XiATXTUDES AD7D DEPARTURES. 


D. 
1 


16i Deg. 


16i Deg. 


16f Deg. 


17 Deg. 


1 
D. S 

1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 1 


Lat. 


Dep. 


.9600 


.2798 


.9588 


.2840 


•9576 


.2882. 


•9563 


.2924 


2 


1. 9201 


•5597, 


1. 9176 


.5680 


1.9151 


•5764: 


1. 9126 


•5847 


2 ; 


3 


2.8801 


.8395. 


2.8765 


.8520 


2.8727 


.86461 


2.8689 


.8771 


3 : 


4 


3.8402 


1.1193, 


3- 8 353 


I-I36I, 


3.8303 


1.152,8} 


3.8252 


1. 1 695 


4 


5 


4.8002 


1. 3991 1 


4-7941 


1. 4201 


4.7879 


1 .4410 


4-78I5 


I.4619 


5 


6 


5.7603 


1.6790 


5-75*9 


I.7041 


5-7454 


I.7292 


5-7378 


I.7542 


6 


7 


6.7203 


1.9588 


6.7117 


1. 9881 


6.7030 


2.0174 


6.6941 j 


2.0466 


1 


8 


7.6804 


2.2386 


7.6706 


2.2721 


7.6606 


2.3056 


7.6504 


2.3390 


8 


i 9 


8.6404 


2.5185! 


8.6294 


2.5561 


8.6181 


2.5938] 


8.6067I 


2.6313 


9 


;10 

i 

j 

i 
1 


9.6005 


2.7983 


9.5882 


2.8402 


9-5757 


2.8820 


9-563o| 


2.9237 


10 
1 


73f Deg. 


73 i Deg. 


73i Deg. 


73 Deg. 


17i Deg. 


17 i Deg. 


17f Deg. 


18 Deg. 


.9550 


.2965 


•9537 


.3007 


.9524 


.3049, 


•95" 


.3090 


2 


1. 9100 


•593 1 


1.9074 


.6014 


1.9048 


.6097 


1. 9021 


.6180 


2 


3 


2.8651 


.8896; 


2.8612 


.9021 ■ 


2.8572 


.9146 


2.8532 


.9271 


3 


4 


3.8201 


1. 1862! 


3.8149 


I.2028 


3.8096 


I.2195 


3.8042 


1. 2361 


4 


5 


4-775 1 


1.4827 j 


4.7686 


i-5°35 


4.7620 


i-5 2 43 . 


4-7553 


I.5451 


5 


6 


5-73 01 


1.779*1 


5.7223 


1.8042; 


5-7I44 


1.8292 


5.7063 


1. 8541 


6 1 


7 


6.6851 


2.0758; 


6.6760 


2.1049 


6.6668 


2.1341! 


6.6574 


2.1631 


7 | 


8 


7.6402 


2.3723 


7.6297 


2.40561 


7.6192 


2.4389 


7.6085 


2.4721 


8 


9 


8.5952 


2.6689! 


8.5835 


2.7064 ' 


8.5716 


2.7438 


8-5595 


2.7812 


9 ! 


10 
1 


9.5502 


2,-9654 j 


9.5372 


3.0071 ; 


9.5240 


3.0486 1 


9.5106 


3.0902 


10 


72f Deg. 


T2i Deg. 


72i Deg. 


72 Deg. 


i 
1 i 


18 i Deg. 


■18i Deg. 


18f Deg. 


19 Deg. 


•9497 


•3 I 3 2 


.9483 


•3 x 73l 


•9469 


.3214 


•9455 


•3*56 


2 


1.8994 


.6263 


1.8966 


.6346 


1.8939 


.6429 


! 1. 8910 


.6511 


2 1 


3 


2.8491 


•9395 


2.8450 


•95191 


2.8408 


.9643 


I 2.8366 


.9767 


3 ! 


4 


3.7988 


1.2527 


3-7933 


1.2692 , 


3.7877 


1.2858 


3.7821 


I.3023 


4 


5 


4-74 8 5 


1.5658 


4.7416 


1.5865 


4-7347 


1.6072 


4.7276 


I.6278 


5 


6 


5.6982 


1.8790 


5.6899 


1.9038 


5.6816 


1.9286 


5.6731 


1-9534 


6 ! 


7 


6.6479 


2.1921 


6.6383 


2.2211 


6.6285 


2.2501 


; 6.6186 


2.2790 


7 


2 


7.5976 


*-5°53 


7.5866 


2.5384 


7-5754 


2.5715 


i 7.5641 


2.6045 


8 


, 9 


8-5473 


2.8185 


8-5349 


2.8557 


8.5224 


2.8930 


8.5097 


2.9301 


9 I 


10 

1 

1 


9.4970 


3.1316 


9.4832 


3.1730 


9.4693 


3.2144 


! 9.4552 


3-*557 


10; 

1 

1 


71| Deg. 


71* Deg. 


! 7H Deg. 




71 Deg. 


19 £ Deg. 


19 i Deg. 


19f Deg. 


20 Deg. 


.9441 


•3*97 


.9426 


.3338 


.9412 


•3379 


•9397 


.3420 


2 


1.8882 


.6594 


1.8853 


.6676 


1.8824 


•6758 


1.8794 


.6840 


2 


3 


2.8323 


.9891 


2.8279 


1. 0014 


2.8235 


1.0138 


i 2.8191 


1. 0261 


3 


4 


3-77 6 4 


1. 3188 


3.7706 


I-335 2 


3-7647 


I-35I7 


1 3-7588 


1. 3681 


4 


5 


4.7204 


1.6485 


4.7132 


1.6690 


4-7°59 


1.6896 


| 4-6985 


1.7101 


5 


6 


5.6645 


1.9781 


5-6558 


2.0028 


5.6471 


2.0275 


1 5.6382 


2.0521 


6 


7 


6.6086 


2.3078 


6.5985 


2.3366 


6.5882 


2.3654 


6.5778 


2.3941 


< 


8 


7-55*7 


2-6375 


7.541 1 


2.6705 


7-5*94 


2.7033 


! 7-5 I 75 


2.7362 


8 


9 


8.4968 


2.9672 


8.4838 


3.0043 


8.4706 


3-°4!3 


1 8.4572 


3.0782 


9 


10 


9.4409 


3.2969 


9-4*64 


3-338i 
Lat. 


j 9.41 1 8 

■ 


3-379* 


. 9-3969 


3.4202 


10 

D. 


D. 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


70| Deg. 


7(H Deg. 


1 70i Deg. 


70 Deg. 



LATITUDES AEfD DEPARTURES. 


D. 

! i 

2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 
7 
8 
9 
10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


202- Deg. 


20£ Deg. 


20f Deg. 


21 Deg. 


D. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


.9382 
I.8764 
2.8146 
3-7528 
4.6910 

5.6291 
6.5673 

7-5°55 
8-4437 
9.3819 


.3461 

.6922 

I.0384 

I-3845 
I.7306 

2.0767 
2.4228 
2.7689 
3.1151 
3.4612 


•9367 
I.8733 
2.8100 

3-7467 
4.6834 

5.6200 
6.5567 

7-4934 
8.4300 
9.3667 


.3502 

.7004 
I.0506 
1.4008 
1.7510 

2.1012 

2-45I5 
2.8017 

3- J 5i9 

3.5021 


•9351 

I.8703 
2.8054 
3-74°5 
4-6757 

5.6108 
6.5459 

7.48 1 1 
8.4162 
9-35I4 


•3543 

.7086 

1.0629 

1.4172 

1.7715 

2.1257 
2.4800 
2.8343 
3.1886 
3-5429 


.9336 
I.8672 
2.8007 

3-7343 
4.6679 

5.6015 

6-5351 
7.4686 
8.4022 
9-3358 


.3584 

.7167 

I.0751 

1-4335 
1.7918 

2.1502 
2.5086 
2.8669 
3-2253 
3-5837 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


69| Deg. 


69 1 Deg. 


69 \ Deg. 


69 Deg. 


21 1 Deg. 


21JDeg. 


21f Deg. 


22 Deg. 


•9320 
1.8640 
2.7960 
3.7280 
4.6600 

5.5920 
6.5241 
7.4561 
8.3881 
9.3201 


.3624 

.72491 

I.0873 

I.4498 

1. 8122 

2.1746 

2.5371 

2.89951 

3.2619 

3.6244 


•9304 
1.8608 
2.7913 

3.7217 
4.6521 

5-5825 
6.5129 

7-4433 
8.3738 
9.3042 


•3665 

•733° 
1.0995 
1.4660 
1.8325 

2.1990 

2-5655 
2.9320 
3.2985 
3.6650 


.9288 
1.8576 
2.7864 

3-7152 
4.6440 

5-5729 
6.5017 

7-43°5 
8-3593 
9.2881 


.3706 

.7411 

1.1117 

1.4822 

1.8528 

2.2233 

2-5939 
2.9645 

3-335° 
3.7056 


.9272 
1.8544 
2.7816 
3.7087 
4.6359 

5-5631 
6.4903 

7-4175 
8.3447 
9.2718 


•3746 

•7492 

1. 1238 

1.4984 

1.8730 

2.2476 
2.6222 
2.9969 

3-37I5 
3.7461 


68f Deg. 


68£ Deg. 


68 \ Deg. 


68 Deg. 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

D. 


22 \ Deg. 


22 i Deg. 


22f Deg. 


23 Deg. 


•9*55 
1.8511 

2.7766 

3.7022 

4.6277 

5-553^ 
6.4788 
7.4043 
8.3299 
9-2-554 


.3786 
•7573 

*- I 359 

1. 5146 

1.8932 

2.2719 
2.6505 
3.0292 
3.4078 
3.7865 


•9239 

1.8478 
2.7716 

3-6955 
4.6194 

5-5433 
6.4672 
7.3910 
8.3149 
9.2388 


.3827 

•7654 
1.1481 

i-53°7 
I-9I34 
2.2961 
2.6788 
3.0615 
3.4442 
3.8268 


.9222 
1.8444 
2.7666 
3.6888 
4.6110 

5-5332 
6.4554 
7.3776 
8.2998 
9.2220 


.3867 

•7734 
1.1601 
1.5468 
I-933 6 

2.3203 

2.7070 
3.0937 
3.4804 
3.8671 


.9205 

1. 8410 
2.7615 
3.6820 
4.6025 

5-523° 
6-4435 
7.3640 
8.2845 
9.2050 


•39°7 

•7815 
1. 1722 
1.5629 
1-9537 
2.3444 

2-735 1 
3-1258 
3.5166 

3-9°73 


671 Deg. 


67 \ Deg. 


67i Deg. 


67 Deg. 


23 i- Deg. 


23 } Deg. 


23f Deg. 


24 Deg. 


.9188 
1.8376 
2.7564 
3.6752 
4.5940 

5.5127 

6-43 1 5 
7-35°3 
8.2691 
9.1879 


•3947 

•7895 
1. 1842 
1.5790 
1-9737 

2.3685 
2.7632 
3.1580 
3-5527 
3-9474 


.9171 

1. 8341 
2.7512 
3.6682 
4-5853 

5.5024 
6.4194 

7-3365 
8.2535 

9.1706 


•3987 

•7975 

1. 1962 

1.5950 

1.9937 

2.3925 
2.7912 
3.1900 
3-5887 
3-9875 


•9 I 53 

1.8306 
2.7459 
3.6612 
4.5766 

5.4919 

6.4072 
7.3225 
8.2378 
9.1531 


.4027 

.8055 

1.2082 

1.6110 

2.0137 

2.4165 
2.8192 
3.2220 
3.6247 
4.0275 


•9 J 35 

1. 8271 
2.7406 
3-6542 
4-5677 

5.4813 
6.3948 
7.3084 
8.2219 
9- I 355 


.4067 

.8135 

1.2202 

1.6269 

2.0337 

2.4404 
2.8472 

3-2539 
3.6606 
4.0674 


D. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


66| Deg. 


| 66* Deg. 


66 \ Deg. 


66 Deg. 



10 



_ _ = — =a 

LATITUDES^ AIT D BEFAET¥EES. 




D. 

1 
2 
3 
4 

5 

6 

7 

8 

9 

10 

i 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 
7 
8 
9 
10 


24£ Deg. 


24J Deg. 


24| Deg. 


25 Deg. 


1 
D. 

1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




.9118 

I.8235 

2 -7353 
3.6470 

4.5588 

5.4706 
6.3823 
7.2941 
8.2059 
9.1176 


.4107 

.8214 

I.2322 

1.6429 

2.0536 

2.4643 
2.8750 
3.2858 
3.6965 
4.1072 


.9100 
1. 8199 
2.7299 
3.6398 
4.5498 

5-4598 
6.3697 
7.2797 
8.1897 
9.0996 


.4147 

.8294 

1. 2441 

1.6588 

2.0735 

2.4882 
2.9029 

3-3*75 
3.7322 
4.1469 


.9081 
1. 8163 

2.7244 
3.6326 
4.5407 

5.4489 
6.3570 
7.2651 

8.1733 
9.0814 


.4187 

•8373 
I.2560 
I.6746 
2.0933 

2.5120 
2.9306 

3-3493 
3.7679 
4.1866 


.9063 
1. 8126 
2.7189 
3.6252 
4-53I5 

5-4378 
6.3442 

7-2505 
8.1568 
9.0631 


.4226 

.8452 

I.2679 

I.6905 

2.II31 

2-5357 
2.9583 
3.3809 
3.8036 
4.2262 




65f Deg. 


65 J Deg. 


65 1 Deg. 


65 Deg. 




25i Deg. 


25J Deg. 


25f Deg. 


26 Deg. 




•9045 
1.8089 

2.7*34 
3.6178 
4.5223 

5.4267 
6.3312 
7.2356 
8.1401 
9.0446 


.4266 

.8531 

I.2797 

1.7063 

2.1328 

2.5594 

2.9860 

3-4I25 

3- 8 39* 
4.2657 


.9026 
1.8052 
2.7078 
3.6103 
4.5129 

5-4155 
6.3181 
7.2207 
8.1233 
9.0259 


•4305 

.8610] 

1.2915 

1.7220 

2.1526 

2.5831 
3.0136 

3-444 1 
3.8746 

4.3051 


.9007 
1. 8014 
2.7021 
3.6028 
4-5035 
5.4042 
6.3049 
7.2056 
8.1063 
9.0070 


•4344 
.8689 

i-3°33 

I-7378 
2.1722 

2.6067 
3.041 1 

34756 
3.9100 

4-3445 


.8988 
1.7976 
2.6964 

3-5952 
4.4940 

5.3928 
6.2916 
7.1904 
8.0891 
8.9879 


•4384 

.8767 

I-3I5 1 

J-7535 

2.1919 

2.6302 
3.0686 
3.5070 

3-9453 
4.3837 




64f Deg. 


64£ Deg. 


64i Deg. 


64 Deg. 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

D. 




26i Deg. 


26* Deg. 


26f Deg. 


27 Deg. 




.8969 

1-7937 
2.6906 

3-5 8 75 
4.4844 

5.3812 

6.2781 
7.1750 
8.0719 
8.9687 


•4423 

.8846 

1.3269 

1.7692 

2.2114 

2.6537 
3.0960 

3-5383 
3.9806 

4.4229 


.8949 
1.7899 
2.6848 
3-5797 
4-4747 
5.3696 
6.2645 

7- J 595 
8.0544 
8.9493 


.4462 

.8924 

1.3386 

1.7848 

2.2310 

2.6772 

3- I2 34 
3.5696 
4.0158 
4.4620 


.8930 
1.7860 
2.6789 

3-57I9 
4.4649 

5-3579 
6.2509 

7-I438 
8.0368 
8.9298 


.4501 

.9002 

1.3503 

1.8004 

2.2505 

2.7006 

3- I 5°7 
3.6008 
4.0509 
4.5010 


.8910 

1.7820 
2.6730 
3.5640 
4.4550 

5.3460 
6.2370 
7.1281 
8.0191 
8.9101 


•454o 

.9080 

1.3620 

1. 8160 

2.2700 

2.7239 
3.1779 
3.6319 
4.0859 

4-5399 




63| Deg. 


63 i Deg. 


63 i Deg. 


63 Beg. 




27i Deg. 


27i Deg. 


271 Deg. 


28 Deg. 




.8890 

1.7780 
2.6671 
3-556i 
4-445 1 

5-334J 
6.2231 
7.1121 

8.0012 
8.8902 


•4579 
•9 I S7 

1.3736 
1.8315 
2.2894 

2.7472 
3.2051 
3.6630 
4.1209 

4-5787 


.8870 
1.7740 
2.6610 
3.5480 

4-435 1 
5.3221 
6.2091 
7.0961 

7-983 1 
8.8701 


.4617 

•9235 

1.3852 
1.8470 
2.3087 

2.7705 
3.2322 
3.6940 

4-1557 
4.6175 


.8850 
1.7700 
2.6550 
3.5400 
4.4249 

5-3°99 
6.1949 
7.0799 
7.9649 
8.8499 


.4656 

.9312 

1.3968 

1.8625 

2.3281 

2.7937 

3-2593 
3.7249 
4.1905 
4.6561 


.8829 
1.7659 
2.6488 
3.5318 

4.4147 

5.2977 
6.1806 
7.0636 
7.9465 
8.8295 


•4695 
.9389 

1.4084 

1.8779 

2.3474 

2.8168 
3.2863 

3-7558 
4.2252 
4.6947 




* 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




62f Deg. 


62* Deg. 


62 i Deg. 


62 Deg. 





11 



LATITUDES AND DEPARTURES, 




D. 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 

i 4 
5 

G 

7 

8 

9 

10 

i 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

I 10 


28 £ Deg. 


28£ Deg. 


28| Deg. 


29 Deg. 


D. 

1 
2 
3 
4 
5 

6 r 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


.8809 
1. 7618 

2.6427 

3-5236 
4.4045 

5-2853 
6.1662 
7.0471 
7.9280 
8.8089 


•4733 
.9466 

1.4200 

1-^933 
2.3666 

2.8399 

3-3 x 3 2 
3.7866 
4.2599 
4.7332 


.8788 

I.7576 
2.6365 

3-5I53 
4.3941 

5.2729 
6.1517 
7.0305 
7.9094 
8.7882 


•4772; 

•95431 

i-43 x 5 
1.9086 

2.3858 

2.8630 
3.3401 

3-8i73 
4.2944 
4.7716 


•8767 
1-7535 
2.6302 
3.5069 
4.3836 

5.2604 
6.1371 

7.0138 
7.8905 
8.7673 


.4810 

.9620 

I.4430 

I.9240 

2.4049 

2.8859 
3.3669 
3.8479 
4.3289 
4.8099 


.8746 

I.7492 
2.6239 

34985 
4-3731 

5-2477 
6.1223 
6.9970 
7.8716 
8.7462 


.4848 

.9696 

I.4544 

I.9392 

2.4240 

2.9089 
3-3937 
3-8785 
4.3633 
4.8481 


61 f Deg. 


61 J Deg. 


61 i Deg. 


61 Deg. 


29 i Deg. 


29 J Deg. 


29f Deg. 


30 Beg. 


.8725 
1.7450 
2.6175 
3.4900 
4.3625 

5.2350 
6.1075 
6.9800 
7.8525 
8.7250 


.4886 

.9772 

1.4659 

1-9545 
2.4431 

2.9317 

3.4203 
3.9090 
4.3976 
4.8862 


.8704 
1.7407 
2.6111 
3.4814 
4-35i8 

5.2221 
6.0925 
6.9628 
7.8332 
8.7036 


.4924 
.9848 

1-4773 
1.9697 

2.4621 

2.9545 

3.4470 

3-9394 
4.4318 
4.9242 


.8682 
1.7364 
2.6046 
3.4728 
4.3410 

5.2092 
6.0774 
6.9456 
7.8138 
8.6820 


.4962 

•9924 
I.4886 
I.9849 
2.48 1 1 

2.9773 

3-4735 
3.9697 
4.4659 
4.9622 


.8660 
1.7321 
2.5981 
3.4641 
4-33°i 
5.1962 
6.0622 
6.9282 
7.7942 
8.6603 


.5000 
1. 0000 
1.5000 
2.0000 
2.5000 

3.0000 
3.5000 
4.0000 
4.5000 
5.0000 


60f Deg. 


60 i Deg. 


60i Beg. 


60 Deg. 


1 
2 
3 
4 
5 

6 

7 

I 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

D. 


30i Deg. 


30 J Deg. 


30f Deg. 


31 Deg. 


.8638 

1.7277 
2.5915 

3-4553 
4.3192 

5.1830 
6.0468 
6.9107 

7-7745 
8.6384 


.5038 

1.0075 

2.0151 
2.5189 

3.0226 
3.5264 
4.0302 

4-534° 
5.0377 


.8616 

1.7233 
2.5849 

3-4465 
4.3081 

5.1698 
6.0314 
6.8930 

7-7547 
8.6163 


•5°75 
1.0151 
1.5226 
2.0302 
2.5377 

3.0452 
3.5528 
4.0603 
4.5678 
5-0754 


•8594 
1. 7188 
2.5782 

3-4376 
4.2970 

5.1564 
6.0158 
6-8753 

7-7347 
8.5941 


•5"3 

1.0226 

1-5339 

2.0452 

2-5565 
3.0678 

3-579 1 
4.0903 
4.6016 
5.1129 


.8572 

1.7143 
2.5715 
3.4287 
4.2858 

5.1430 
6.0002 
6.8573 

7-7145 
8.5717 


.5150 
1.0301 
1.5451 

2.0602 
2.5752 

3.0902 
3.6053 
4.1203 

4-6353 
5^504 


59| Deg. 


591 Deg. 


59 i Deg. 


59 Deg. 


3U Deg. 


311 Deg. 


31f Deg. 


32 Deg. 


.8549 
1.7098 
2.5647 
3.4196 
4.2746 

5-1295 
5.9844 
6.8393 
7.6942 
8.5491 


.5188 
i-o375 
I-5563 
2.0751 

2-5939 
3.1126 
3.6314 
4.1502 
4.6690 
5.1877 


.8526 
1.7053 

2-5579 
3.4106 
4.2632 

5.1158 
5.9685 
6.8211 
7.6738 
8.5264 


.5225 
1.0450 
1.5675 
2.0900 
2.6125 

3-i35o 

3-6575 
4.1800 
4.7025 
5.2250 


.8504 
1.7007 
2.5511 
3.4014 
4.2518 

5.1021 

5-9525 

6.8028 

1 7.6532 

1 8.5035 

1 


.5262 
1.0524 
1.5786 
2.1049 
2.6311 

3-J573 
3-6835 
4.2097 

4-7359 
5.2621 

Lat. 


.8480 
1. 6961 

2-5441 
3.3922 
4.2402 

5.0883 

5-9363 
6.7844 
7.6324 
8.4805 


•5299 
1.0598 
1.5898 
2.1197 
2.6496 

3- I 795 

3.7094 

4-2394 
4.7693 
5.2992 


D. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 


58| Deg. 


58 * Deg. 


1 581- Deg. 


58 Deg. 



12 



LATITUDES AND DEPARTURES, 




D. 

1 

i 2 
3 

1 4 
5 

! 6 
7 
8 
9 

! io 
1 

2 
3 
4 

5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


32 i Deg. 


32J Deg. 


32| Deg. 


33 Deg. 


D. | 

1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




•8457i 
I.6915 

2.5372 
3-3829 
4.2286 

5.0744 
5.9201 
6.7658 
7.6116 
8-4573 


.5336 
I.0672 
I.6008 
2.1345 
2.6681 

3.2017 

3-7353 
4.2689 
4.8025 
5-336i 


•8434 
1.6868 
2.5302 
3.3736 
4.2170 

5.0603 

5-9°37 
6.7471 

7-59°5 
8-4339 


•5373 
1.0746 
1.6119 1 
2.1492 
2.6865 

3.2238 

3-76ii 
4.2984 

4-8357 
5-373° 


.8410 
1. 6821 
2.5231 
3.3642 
4.2052 

5.0462 
5.8873 
6.7283 
7.5694 
8.4104 


.5410 
1. 0819 
I.6229 
2.1639 
2.7049 

3-2458 
3.7868 
4.3278 
4.8688 
5.4097 


.8387 
1.6773 
2.5160 

3-3547 
4-1934 
5.0320 
5.8707 
6.7094 
7.5480 
8.3867 


•5446 
1.0893 
I.6339 
2.1786 
2.7232 

3.2678 
3.8125 

4-3571 
4.9018 
5.4464 




571 Deg. 


57i Deg. 


57i Deg. 


57 Deg. 




. 33i Deg. 


33 i Deg. 


33f Deg. 


34 Deg. 




.8363 

1.6726 
2.5089 

3-3451 
4.1814 

5- OI 77 
5.8540 
6.6903 
7.5266 
8.3629 


•5483 
1.0966 
1.6449 
2.1932 
2.7415 

3.2898 
3.8381 
4.3863 
4.9346 
5.4829 


•8339 
1.6678 
2.5017 

3-3355 
4.1694 

5-o°33 
5-8372 
6.6711 
7.5050 
8.3389 


•55*9 
1.1039 

1-6558 

2.2077 
2.7597 

3.3116 

3.8636 

4-4I55 
4.9674 

5.5194 


.8315 

1.6629 
2.4944 
3-3259 
4- J 573 
4.9888 
5.8203 
6.6518 
7.4832 
8.3147 


•5556 
I.IIII 

I.6667 
2.2223 
2.7779 

3-3334 
3.8890 
4.4446 
5.0001 

5-5557 


.8290 
1. 6581 
2.4871 
3.3162 
4.1452 

4.9742 
5-8033 
6.6323 
7.4613 
8.2904 


•5592 
1.1184 
I.6776 
2.2368 
2.7960 

3-3552 
3-9*44 
4-4735 
5.0327 

5-59*9 




56f Deg. 


56J Deg. 


56i Deg. 


56 Deg. 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 
8 

9 
10 




34i Deg. 


34£ Deg. 


34| Deg. 


35 Deg. 




.8266 
1.6532 
2.4798 
3.3064 
4.1329 

4-9595 
5.7861 
6.6127 

7-4393 
8.2659 


.5628 
1. 1256 
1.6884 
2.2512 
2.8140 

3-3768 
3.9396 

4.5024 
5.0652 
5.6280 


.8241 
1.6483 
2.4724 
3.2965 
4.1206 

4.9448 
5.7689 
6.5930 

7-4171 
8.2413 


.5664 
1. 1328 
1.6992 
2.2656 

2.8320 

3-3984 
3.9648 

4-53 12 
5.0977 
5.6641 


.8216 

1.6433 
2.4649 
3.2866 
4.1082 

4.9299 

5-75I5 
6.5732 
7.3948 
8.2165 


.5700 
1. 1400 
1. 7100 
2.2800 
2.8500 

3.4200 
3.9900 

4.560O 
5.I3OO 

5.7OOO 


.8192 
1.6383 

2-4575 
3.2766 
4.0958 

4.9149 

5-7341 
6.5532 
7.3724 
8.1915 


•5736 
1-1472 
1.7207 
2.2943 
2.8679 

3-44I5 
4.0150 
4.5886 
5.1622 
5-7358 




55 1 Deg. 


55 i Deg. 


55? Deg. 


55 Deg. 




35 \ Deg. 


35J Deg. 


35| Deg. 


36 Deg. 




.8166 
1.6333 
2.4499 
3.2666 
4.0832 

4.8998 
5.7165 
6.5331 

7-3498 
8.1664 


•577i 

I-I543 

i 1-73*4 

2.3086 

2.8857 

3.4629 
4.0400 
4.6172 

5-1943 

5-77I5 


.8141 
1.6282 
2.4423 

3-2565 
4.0706 

4.8847 
5.6988 
6.5129 
7.3270 
8.1412 


•5807 
1 1.1614 

i-742i 
2.3228 
2.9035 

3.4842 
4.0649 
4.6456 
5.2263 
5.8070 

Lat. 


.8116 

1. 6231 

2-4347 
3.2463 

4-°579 
4.8694 
5.6810 
6.4926 
7.3042 
j 8.1157 


.5842 
1. 1685 
1.7527 
2.337O 
2.9212 

3-5055 
4.0897 

4.674O 

5.2582 

5-8425 


.8090 
1. 6180 
2.4271 
3.2361 
4.0451 

4.8541 
5.6631 
6.4721 
7.2812 
8.0902 


.5878 

1-1756 
1.7634 
2.3511 
2.9389 

3.5267 

4- "45 
4.7023 
5.2901 

5-8779 




D. 


Dep. 


Lat. 

1 


Dep. 


Dep. 


Lat. 

1 


! 

Dep. 


Lat. 




54f Deg. 


54£ Deg. 


; 54 \ Deg. 


54 Deg. 





13 



LATITUDES AND DEPARTURES. 


D. 

1 
2 
3 
4 
5 

6 

7 

8 

: 9 

1 10 

l 

2 
3 
4 
5 

6 

7 

8 

9 

10 

1 

1 1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


361 Deg. 


36* Deg. 


36f Deg. 


37 Deg. 


D. 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


Lat. 


Dep. 

•5913 
1. 1826 

1-7739 
2.3652 
2.9565 

3-5479 
4.1392 
4.7305 
5.3218 
5.9131 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


.8064 
1. 6129 
2.4193 
3.2258 
4.0322 

4.8387 

5-645I 
6.4516 

7.2580 

8.0644 


.8039 
I.6077 
2.4116 

3-2154 
4.0193 

4.8231 
5.6270 
6.4309 

7-2347 
8.0386 


•5948 
1. 1896 
I.7845 
2.3793 
2.9741 

3-5689 
4.1638 
4.7586 

5-3534 
5.9482 


.8013 

I.6025 
2.4038 
3.2050 
4.0063 

4.8075 
5.6088 
6.4100 
7-2II3 
8.0125 


.5983 
1. 1966 
I.7950 

2-3933 
2.9916 

3-5899 
4.1883 
4.7866 
5.3849 
5.9832 


.7986 
1-5973 
2-3959 
3-1945 
3.9932 

4.7918 

5-59°4 
6.3891 

7.1877 
7.9864 


.6018 
I.2036 
I.8054 
2.4073 
3.0091 

3.6109 

4.2127 
4.8145 
5.4163 
6.0181 


53f Deg. 


53 J Deg. 


531 Deg. 


53 Deg. 


371 Deg. 


37* Deg. 


37f Deg. 


38 Deg. 


.7960 
1.5920 
2.3880 
3.1840 
3.9800 

4.7760 
5-5720 
6.3680 
7.1640 
7.9600 


.6053 
1. 2106 
1. 8159 
2.4212 
3.0265 

3.6318 

4-237I 
4.8424 

54476 
6.0529 


•7934 
1.5867 
2.3801 
3.1734 
3.9668 

4.7601 

5-5535 
6.3468 
7.1402 
7-9335 


.6088 
1.2175 
1.8263 
2.4350 
3.0438 

3.6526 
4.2613 
4.8701 
5.4789 
6.0876 


.7907 
1. 5814 
2.3721 
3.1628 
3-9534 
4.7441 

5-5348 
6.3255 
7.1162 
7.9069 


.6122 
1.2244 
1.8367 
2.4489 
3.0611 

3-6733 
4.2855 
4.8977 
5.5100 
6.1222 


.7880 
1.5760 
2.3640 
3.1520 
3.9401 

4.7281 
5-5i6i 
6.3041 
7.0921 
7.8801 


.6157 

I-23I3 

I.8470 
2.4626 
3- 783 
3.6940 
4.3096 

4-9253 
5-54IO 
6.1566 


52| Deg. 


52* Deg. 


521 Deg. 


52 Deg. 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

D. 


381 Deg. 


38* Deg. 


38 f Deg. 


39 Deg. 


•7853 
1.5706 

2.3560 

3-HI3 
3.9266 

4.71 19 

5.4972 
6.2825 
7.0679 
7.8532 


.6191 
1.2382 
1.8573 
2.4764 
3-°955 

3-7I46 
4-3337 
4.9528 
5.5718 
6.1909 


.7826 
1.5652 
2.3478 
3.1304 
3.9130 

4.6956 

5-4783 
6.2609 

7-0435 
7.8261 


.6225 
1.2450 
1.8675 
2.4901 
3.1126 

3-735 1 
4-3576 
4.9801 
5.6026 
6.2251 


•7799 
1.5598 

2.3397 

3-H95 

3.8994 

4-6793 
5.4592 
6.2391 
7.0190 
7.7988 


.6259 
1.2518 

1.8778 
2.5037 
3.1296 

3-7555 
4-38i5 
5.0074 

5-6333 
6.2592 


.7771 

1-5543 
2-33H 
3.1086 
3.8857 

4.6629 
5.4400 
6.2172 
6-9943 
7-77I5 


.6293 
I.2586 
I.8880 

2-5I73 
3.1466 

3-7759 
4.4052 
5.0346 
5.6639 
6.2932 


51| Deg. 


51* Deg. 


511 Deg. 


51 Deg. 


39* Deg. 


39* Deg. 


39f Deg. 


40 Deg. 


•7744 
1.5488 
2.3232 
3.0976 
3.8720 

4.6464 
5.4207 
6.1951 
6.9695 

7-7439 


•6327 
1.2654 
1. 8981 
2.5308 
3.1635 

3.7962 
4.4289 
5.0616 
5.6943 
6.3271 


.7716 
1.5432 
2.3149 
3.0865 
3-8581 

4.6297 
5.4014 
6.1730 
6.9446 
7.7162 


.6361 
1.2722 
1.9082 

2-5443 
3.1804 

3.8165 
4.4525 
5.0886 

5-7247 
6.3608 


.7688 

1-5377 
2.3065 

3-°754 
3.8442 

4.6131 

5-38i9 
6.1507 
6.9196 
7.6884 


•6394 
1.2789 
1. 9183 
2.5578 
3.1972 

3.8366 
4.4761 
5-"55 
5-755° 
6.3944 


.7660 
1.5321 
2.2981 
3.0642 
3.8302 

4.5963 
5.3623 
6.1284 
6.8944 
7.6604 


.6428 
1.2856 
1.9284 
2.5712 
3.2139 

3-8567 
4.4995 
5.1423 
5.7851 
6.4279 


L 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


50f Deg. 


50* Deg. 


501 Deg. 


50 Deg. 



14 



! 



LATITUDES AETO DEPARTURES. 


1 


D. 

1 

1 2 

1 3 

4 

5 

6 

7 

8 

9 

10 

1 

2 

! I 

5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

i 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


40£ Deg. 


40J Deg. 


40| Deg. 


41 Deg. 


D. 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




.7632 
1.5265 
2.2897 
3.0529 
3.8162 

4-5794 
5.3426 
6.1059 
6.8691 
7.6323 


.6461 
I.2922 
I.9384 
2.5845 
3.2306 

3.8767 
4.5229 
5.1690 

5.8I5I 
6.4612 


.7604 
I.5208 
2.2812 
3.0416 
3.8020 

4.5624 
5.3228 
6.0832 

6.8437 
7.6041 


•6494 
I.2989 
I.9483 
2.5978 
3.2472 

3.8967 
4.5461 

5-I956 
5.8450 
6.4945 


•7576 
I.5151 

2.2727 
3.0303 
3-7878 
4.5454 

5-3030 
6.0605 

6.8181 

7-5756 


.6528 
L3055 
I-9583 
2.6110 
3.2638 

3.9166 

4-5693 
5.2221 
5.8748 
6.5276 


•7547 
1.5094 
2.2641 
3.0188 

3-7735 
4.5283 
5.2830 
6.0377 
6.7924 
7.5471 


.6561 
I.3121 
I.9682 
2.6242 
3.2803 

3-9364 
4.5924 
5.2485 
5.9045 
6.5606 




49f Deg. 


49 \ Beg. 


49 i Deg. 


49 Deg. 




41 1 Deg. 


41 £ Deg. 


41 f Deg. 


42 Beg. 




.7518 

1.5037 
2.2555 
3.0074 
3-7592 

4,5110 

5.2629 
6.0147 
6.7666 
7.5184 


•6593 
I.3187 
I.9780 
2.6374 
3.2967 

3.9561 
4.6154 
5.2748 
5-9341 
6-5935 


.7490 
1.4979 
2.2469 
2.9958 
3.7448 

4-4937 
5.2427 
5.9916 
6.7406 
7.4896 


.6626 
I.3252 
I.9879 
2.6505 

3-9757 
4.6383 
5.3010 
5.9636 
6.6262 


.7461 
1.4921 
2.2382 
2.9842 

3-73°3 

4-4763 
5.2224 
5.9685 
6.7145 
7.4606 


.6659 
I-33I8 
I.9976 
2.6635 

3-3294 

3-9953 
4.6612 

5-327I 
5.9929 
6.6588 


•743 1 
1.4863 
2.2294 
2.9726 
3-7I57 
4.4589 
5.2020 
5.9452 
6.6883 
743 J 4 


,6691 

I-3383 
2.0074 
2.6765 

3-3457 
4.0148 
4.6839 

5-353° 
6.0222 
6.6913 




48f Deg. 


48 h Deg. 


48 1 Deg. 


48 Deg. 


1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

1 

3 
4 
5 

6 

7 

8 

9 

10 

D. 




42 \ Deg. 


42£ Deg. 


42f Deg. 


43 Deg. 




.7402 
1.4804 
2.2207 
2.9609 
3-7011 

4-4413 
5-1815 
5.9217 
6.6620 
7.4022 


.6724 
I.3447 
2.0171 
2.6895 
3.3618 

4.0342 
4.7066 

5-3789 
6.0513 
6.7237 


•7373 
1.4746 
2.2118 
2.9491 
3.6864 

4.4237 
5.1609 
5.8982 
6.6355 
7.3728 


•6756 
1. 3512 
2.0268 

2.7024 
3.3780 

4-°535 
4.7291 

5-4047 
6.0803 

6-7559 


•7343 
1.4686 
2.2030 

2-9373 
3.6716 

4.4059 

5- J 403 
5.8746 
6.6089 
7.3432 


.6788 

I-3576 
2.0364 
2.7152 
3.3940 

4.0728 
4.7516 

5-43°4 
6.1092 
6.7880 


•73 J 4 
1.4627 
2.1941 
2.9254 
3.6568 

4.3881 
5.1195 
5.8508 
6.5822 
7-3J35 


.6820 
1.3640 
2.0460 
2.7280 
3.4100 

4.0920 
4.7740 
5.4560 
6.1380 
6.8200 




471 Deg. 


47£ Deg. 


47i Deg. 


47 Deg. 




43 £ Deg. 


43 £ Deg. 


43f Deg. 


44 Deg. 




.7284 
1.4567 
2.1851 
2.9135 
3.6419 

4.3702 
5.0986 
5.8270 

6-5553 
7.2837 


.6852 

I-3704 
2.0555 
2.7407 
34259 

4.1IH 

4.7963 

5-48I5 
6.1666 

6.8518 


•7254 
1.4507 
2.1761 
2.9015 
3.6269 

4.3522 
5.0776 
5.8030 
6.5284 

7-2537 


.6884 
1.3767 
2.0651 

2-7534 
3.4418 

4.1301 
4.8185 
5.5068 
6.1952 
6.8835 


.7224 
1.4447 
2.1671 
2.8895 
3.6118 

4.3342 
5.0565 
5-7789 
6.5013 
7.2236 


.6915 
1.3830 
2.0745 
2.7661 
34576 

4.1491 

4.8406 

5.5321 
6.2236 
6.9151 


•7193 

1.4387 

2.1580 
2.8774 
3.5967 
4.3160 

5-°354 
5-7547 
6.4741 

7-1934 


.6947 

1-3893 
2.0840 

2.7786 

3-4733 

4.1680 

4.8626 

5-5573 
6.2519 
6.9466 




D. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




46f Deg. 


46£ Deg. 


46£ Deg. 


46 Deg. 





15 



LATITUDES AWTD DEPARTURES. 


D. 

J 1 
2 
3 
4 
5 

6 

7 

8 

9 

10 


44 i Deg. 


44| Deg. 


44f Deg. 


45 Deg. 


D. 

1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

D. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


.7163 

1.4326 
2.1489 
2.8652 
3-58I5 

4.2978 

5.0141 

5-73°4 
6.4467 
7.1630 


.6978 
I.3956 
2.0934 
2.7912 
3.4890 

4.1867 
4.8845 
5-5823 
6.2801 
6.9779 


•7133 

I.4265 

2.1398 
2.8530 
3.5663 

4.2795 
4.9928 
5.7060 
6.4193 
7.1325 


.7009 
1. 4018 
2.1027 
2.8036 
3-5°45 
4.2055 
4.9064 
5.6073 
6.3082 
7.0091 


.7102 

I.4204 
2.1306 
2.8407 

3-55°9 
4.2611 

4-97I3 
5.6815 
6.3917 
7.1019 


.7040 
1.4080 
2.1120 
2.8161 
3.5201 

4.2241 
4.9281 
5.6321 
6.3361 
7.0401 


.7071 
I.4142 
2.1213 
2.8284 
3-5355 
4.2426 

4-9497 
5.6569 
6.3640 
7.0711 


.7071 
1.4142 
2.1213 
2.8284 
3-5355 
4.2426 
4.9497 
5.6569 
6.3640 
7.071 1 


D. 


Dep. 


Lat. 


• Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


45f Deg. 


45£ Deg. 


451 Deg. 


45 Deg. 



TABLE OF USEFUL NUMBERS. 

Logarithms. 
Ratio of circumference to diameter n = 3.1415926536 0.4971499 

Area of circle to radius 1 = " " 

Surface of sphere to diameter 1 = " " 

Area of circle to diameter 1 = 7853981634 — 1.8950899 

Base of Napierian Logarithms = 2.7182818285 4342945 

Modulus of common " = 4342944819 — 1.6377843 

Equatorial radius of the earth, in feet = 20923599.98 7.3206364 

Polar " " " = 20853657.16 7.3191823 

Length of seconds pendulum, in London, in inches = 39-13929. 

" " « Paris " = 39-1285. 

" " " New York " =39.1012. 

U. S. standard gallon contains 231 c. in., or 58372.175 grains = 8.338882 lbs. avoir- 
dupois of water at 39. 8° Fahr. 
U. S. standard bushel contains 2150.42 c. in., or 77.627413 lbs. av. of water at 39. 8° 

• Fahr. 
British imperial gallon contains 277.274 c. in., = 1.2003 wine gallons of 231 c. in. 
French metre = 39.37079 in. == 3.28089917 feet. 

" toise = 6.39459252 feet. 

" are = 100 sq. metres = 1076.4299 sq. ft. 

" hectare = 100 ares = 2.471 143 acres = 107642.9936 sq. ft. 

" litre = 1 cubic decimeter = 61.02705 c. in. = .2641 8637 gallons of 231 c. in. 

u hectolitre = 100 litres = 26.418637 gallons. 
I pound avoirdupois = 7000 grs. = 1. 21 5277 pounds Troy. 
1 " Troy = 5760 grs. = .822857 pounds avoir. 
I gramme = 15.442 grains. 

1 kilogramme = 1000 grammes = 15442 grs. = 2.20607 lbs. avoir. 
Tropical year = 365 d. 5 h. 45 m. 47.588 sec. 



16 



TABLE 



or THE 



LOGARITHMS OF NUMBERS, 



FROM 



1 to 10,000. 



17 



A TABLE 



OP THE 



LOGARITHMS OF NUMBERS 



FKOM 1 TO 10,000. 



N. 


Log. 


N. 


Log. 


N. 


lAg. 


N. 


Log. 




1 


o.oooooo 


26 


i-4 x 4973 


51 


1.707570 


76 1 


.880814 




2 


0.301030 


27 


1.431364 


52 


1.716003 


77 1 


.886491 




1 3 


0.477121 


28 


1.447158 


53 


1.724276 


78 1 


.892095 




4 


0.602060 


29 


1.462398 


54 


1.732394 


79 1 


.897627 




5 


0.698970 


30 


1.477121 


55 


1.740363 


80 1 


.903090 




6 


0.778151 


31 


1. 491362 


56 


1. 748188 


81 1 


.908485 




7 


0.845098 


32 


1.505150 


57 


i-755 8 75 


82 1 


.913814 




8 


0.903090 


33 


1.518514 


58 


1.763428 


83 1 


.919078 




9 


0.954243 


34 


i-SlWV 


59 


1.770852 


84 1 


924279 




10 


1. 000000 


35 
36 


1.544068 


60 


1.778151 


85 1 


929419 




11 


1.041393 


1.556303 


61 


1.785330 


86 1 


934498 




12 


1.079181 


37 


1.568202 


62 


1.792392 


87 1 


939519 




13 


1.113943 


38 


1.579784 


63 


I-79934I 


88 1 


944483 




14 


1.146128 


39 


1. 591065 


64 


1. 806180 


89 1 


949390 




15 


1.176091 


40 


1.602060 


65 


1.812913 


90 1 


954H3 




16 


1. 204120 


41 


1. 612784 


66 


1. 819544 


91 1 


959041 




17 


1.230449 


42 


1.623249 


67 


1.826075 


92 1 


963788 




18 


1.255273 


43 


1.633468 


68 


1.832509 


93 1 


968483 




19 


1.278754 


44 


I-643453 


69 


1.838849 


94 1 


973128 




20 


1. 301030 


45 


1. 653213 


70 


1.845098 


95 1 


977724 




21 


I. 322219 


46 


1.662758 


71 


1. 851258 


96 1 


982271 




22 


1.342423 


47 


1.672098 


72 


1.857332 


97 1 


986772 




23 


1. 361728 


48 


1.681241 


73 


1 863323 


98 1. 


991226 




24 


1.380211 


49 


1. 690196 


74 


1 869232 


99 1. 


995635 




25 


1.397940 1 


50 


1.698970 


75 


1. 875061 


100 2. 


000000 





19 



N. 100. LOGARITHMS. Log. 000. 


N. 

100 

101 
102 

103 
104 

105 
106 
107 
108 
109 

110 

111 
112 
113 
114 

115 
116 
117 
118 
119 





1 


2 


3 

1301 

5609 
9876 
4100 
8284 

2428 
6533 
°6oo 
4628 
8620 

2576 

6495 
O380 
4230 
8046 

1829 
558o 
9298 
2985 
6640 

~°a66 

3861 
7426 
0963 

447i 

795i 

1403 

4828 

8227 

1599 

4944 
8265 

1560 
4830 
8076 

1298 

449 6 
7671 

°822 

3951 

7058 

°I42 

3205 

6246 
9266 

2266 

5244 
8203 

1141 

4060 

6959 
9839 

2700 

5542 
8366 

1 171 

3959 
6729 
9481 
2216 


4 5 


6 

2598 
6894 

^47 
5360 

9532 

3664 

7757 

!8l2 

5830 

9811 

3755 
7664 

! 538 
5378 
9185 

2958 
6699 
0407 
4085 
_77 3 i 

J 347 
4934 
8490 

2 oi8 
55i8 
8990 
2434 
5851 
9241 
2605 

5943 
9256 

2544 
5806 
9045 

2260 

545 1 
8618 
1763 
4885 

7985 
^63 

4120 

7 J 54 
°i68 

3161 
6134 
9086 
2019 
4932 
7825 
%99 

3555 
6391 
9209 

2010 
4792 
7556 
°3°3 
3°33 


7 


8 


9 


oooooo 

4321 

8boo 
012837 

7033 

021189 
5306 
9384 

033424 
7426 


o434 

475 1 
9026 

3 2 59 
745 1 
1603 

5715 
9789 
3826 
7825 

1787 

57H 
9606 

34 6 3 
7286 

1075 
4832 

8557 
2250 

59" 

9543 

3144 
6716 
0258 
3772 
7257 

0715 
4146 

7549 
0926 

4277 
7603 
0903 
4178 
7429 

0655 
3858 

7037 
°i94 

33 2 7 

6438 

95 2 7 
2594 
5640 
8664 

1667 
4650 
7613 

0555 
3478 

6381 
9264 
2129 

4975 
7803 

0612 

34°3 
6176 
8932 
1670 

1 . 


0868 
5181 

945i 
3680 
7868 

2016 
6125 

°i95 

4227 
8223 

2182 
6105 

9993 
3846 

7666 

> 1452 
5206 
8928 
2617 
6276 

9904 

3503 

7071 

°6n 
4122 

7604 
1059 

4487 
7888 
1263 

46 1 1 

7934 
1231 

4504 
7753 
0977 
4177 

7354 
0508 

3 6 39 

6748 

9835 
2900 

5943 
8965 

1967 

4947 
7908 
0848 
3769 

6670 

9552 
2415 

5259 
8084 

0892 
3681 

6 453 
9206 

1943 

2 


J 734 
6038 
°3oo 
4521 
8700 

2841 
6942 
'004 
5029 
9017 


2l66 
6466 

°7 2 4 
4940 
9116 

3252 

7350 
1408 

543° 
9414 


3029 
7321 
1570 

5779 
9947 

4°75 
8164 
2216 
6230 

2O7 

4148 

8053 

1924 

5760 

95 6 3 

3333 

7071 
0776 

445 1 
8094 

1707 

5291 
8845 

2 37° 
5866 

9335 

2777 
6191 

9579 

2940 

6276 
9586 
2871 
6131 
9368 

2580 
5769 

8934 
2076 

5i9 6 

8294 

1370 
4424 
7457 

o 4 6 9 

3460 
6430 
9380 
2311 
5222 

~87i3 
° 9 86 

3839 
6674 

9490 

2289 
5069 
7832 
°577 
33°5 

7 


3461 

7748 
1993 
6197 
0361 

4486 
8571 
2 6i 9 
6629 

%02 

4540 

8442 

2309 

6142 

9942 

3709 

7443 
1145 
4816 
8457 
2067 
5647 
9198 
2721 
6215 
9681 

3 IJ 9 

653 1 
9916 

3 2 75 

6608 

99 J 5 

3198 

6456 
9690 

2900 
6086 
9249 

2 389 
55°7 
8603 

] 6 7 6 
4728 

7759 
° 7 6 9 

3758 
6726 
9674 
2603 
5512 

8401 
I272 

4123 
6956 
9771 

2567 

5346 

8107 

0850 

_3577 

8 


3891 
8i74 
2415 
6616 
°775 
4896 
8978 

3 02I 
7028 

o 99 8 


041393 

53 2 3 

9218 

053078 

6905 

060698 

4458 

8186 

071882 

5547 


2969 
6885 
0766 
4613 
8426 

2206 

5953 
9668 

3352 

7004 


3362 

7275 
'i53 
4996 
8805 

2582 
6326 
0038 
37i8 
7368 

0987 
4576 
8136 
'667 
5169 

8644 
2091 

55i° 
8903 
2270 


4932 
8830 
2694 
6524 
0320 

4083 

7815 

1514 
5182 
8819 

2426 

6004 

9552 

3 o7i 
6562 

O26 

3462 
6871 

°253 
3609 


120 
121 
122 
123 
124 

125 
126 
127 
128 
129 

130 
131 
132 
133 
134 

135 
136 
137 
138 
139 


079181 

082785 

6360 

99°5 
093422 

6910 
100371 

3804 
• 7210 
110590 

"3943 
7271 

120574 
3852 
7io5 

i3°334 
3539 

6721 

9879 
I43°i5 
146128 

9219 
152288 

533 6 
8362 

161368 

4353 

7317 

170262 

3186 


°6 2 6 

4219 

778i 

V5 
4820 

8298 
1747 
5169 
8565 
1934 


5278 

8595 
1888 

5I5 6 
8399 
1619 

4814 

7987 
1136 
4263 


5611 
8926 
2216 
5481 
8722 

1939 

5i33 
8 3°3 
*45° 
4574 

7676 
0756 

3815 
6852 
9868 

2863 

5838 
8792 
1726 
4641 


6940 
0245 

3525 
6781 

°OI2 

3219 

6403 
9564 
2702 
5818 

89II 
^82 
5032 
8o6l 
I068 

4055 
7022 
9968 
2895 
5802 

8689 
I558 
4407 
7239 

°o5i 

2846 
5623 
8382 

] I24 

3848 


140 
141 
142 
143 
144 

145 

146 
147 
148 
149 


7367 
0449 

35io 
6 549 
9567 

2564 
554i 
8497 
1434 

435i 


150 
151 
152 
153 
154 

155 
156 
157 
158 
159 


176091 
8977 

1 81 844 
4691 
7521 

190332 
3125 
5900 
8657 

201397 


7248 

°I26 

2985 
5825 
8647 

1451 

4237 
7005 

9755 
2488 


753 6 
V3 
3270 
6108 
8928 

1730 

45*4 
7281 

O29 

2761 


N. 





3 


4: 


5 


6 


9 



20 



N. 


L60. LOGARITHMS 




Log. 204. i 


N. 





1 


2 


3 


4 


5 


6 

5746 


7 
6016 


8 

"6286 


9 1 


160 


204120 


439 1 


4663 


4934 


5204 


5475 


6556 


161 


6826 


7096 


7365 


7 6 34 


7904 


8173 


8441 


8710 


8979 


9 2 47 


162 


9515 


9783 


0051 


°3 J 9 


0586 


°853 


J I2I 


'388 


'654 


! 92I i 


163 


212188 


2454 


2720 


2986 


3252 


35i8 


3783 


4049 


43 J 4 


4579 ! 


164 


4844 


5109 


5373 


5638 


5902 


6166 


643O 


6694 


6957 


7221 | 


165 


7484 


7747 


8010 


8273 


8536 


8798 


9060 


9323 


9585 


9846 


166 


220108 


0370 


0631 


0892 


1153 


1414 


1675 


1936 


2196 


2456 


167 


2716 


2976 


3236 


3496 


3755 


4015 


4274 


4533 


4792 


5051 


168 


53°9 


55^8 


5826 


6084 


6342 


6600 


6858 


7115 


7372 


7630 


169 


7887 


8144 


8400 


8657 


8913 


9170 


9426 
1979 


9682 


9938 
2488 


°i93 


170 


230449 


0704 


0960 


1215 


1470 


1724 


2234 


2742 


171 


2996 


3250 


3504 


3757 


401 1 


4264 


45*7 


4770 


5023 


5276 


172 


55*8 


5781 


6033 


6285 


6537 


6789 


7041 


7292 


7544 


7795 


173 


8046 


8297 


8548 


8799 


9049 


9299 


955° 


9800 


°o5o 


°3oo 


174 


240549 


0799 


1048 


1297 


1546 


1795 


2044 


2293 


2541 


2790 


175 


3038 


3286 


3534 


3782 


4030 


4277 


4525 


4772 


5019 


5266 


176 


5513 


5759 


6006 


6252 


6499 


6745 


6991 


7237 


7482 


7728 


177 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


0176 


178 


250420 


0664 


0908 


1151 


1395 


1638 


1881 


2125 


2368 


. 2610 


179 


2853 


3096 
55H 


3338 


3580 
5996 


3822 


4064 
6477 


4306 
"6778 


4548' 
6958 


4790 
7198 


5°3 X 


180 


255 2 73 


5755 


6237 


7439 


181 


7679 


7918 


8158 


8398 


8637 


8877 


9116 


9355 


9594 


9833 


182 


260071 


0310 


0548 


0787 


1025 


1263 


1501 


1739 


1976 


2214 


183 


2451 


2688 


2925 


3162 


3399 


3 6 3 6 


3873 


4109 


4346 


4582 


184 


4818 


5°54 


5290 


552-5 


57 6x 


599 6 


6232 


6467 


6702 


6937 


185 


7172 


7406 


7641 


7875 


8110 


8344 


8578 


8812 


9046 


9279 


186 


9513 


9746 


9980 


0213 


0446 


0679 


O912 


I144 


i 377 


x 6o9 


187 


271842 


2074 


2306 


2538 


2770 


3001 


3233 


3464 


3696 


3927 


188 


4158 


4389 


4620 


4850 


5081 


53 11 


5542 


5772 


6002 


6232 


189 


6462 


6692 
8982 


6921 


7151 
9439 


7380 


7609 


7838 

°I23 


8067 
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8296 


8525 


190 


278754 


9211 


9667 


9895 


°8o6 


191 


281033 


1261 


1488 


1715 


1942 


2169 


2396 


2622 


2849 


3°75 


192 


3301 


35*7 


3753 


3979 


4205 


4431 


4656 


4882 


5107 


5332 


193 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7130 


7354 


7578 


194 


7802 


8026 


8249 


8473 


8696 


8920 


9H3 


9366 


9589 


9812 


195 


290035 


0257 


0480 


0702 


0925 


1 147 


1369 


1591 


1813 


2034 


196 


2256 


2478 


2699 


2920 


3H 1 


33 6 3 


3584 


3804 


4025 


4246 


197 


4466 


4687 


4907 


5 I2 7 


5347 


55 6 7 


5787 


6007 


6226 


6446 


198 


6665 


6884 


7104 


73 2 3 


7542 


7761 


7979 


8198 


8416 


8635 


199 


8853 


9071 

1247 


9289 
1464 


9507 
1681 


9725 


9943 
2114 


°i6i 
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u 378 
2547 


u 595 
2764 


°8i3 
2980 


200 


301030 


1898 


201 


3196 


3412 


3628 


3844 


4059 


4275 


449 1 


4706 


4921 


5136 


202 


535i 


5566 


578i 


599 6 


6211 


6425 


6639 


6854 


7068 


7282 


203 


7496 


7710 


7924 


8i37 


8351 


8564 


8778 


8991 


9204 


9417 


204 


9630 


9843 


0056 


°268 


° 4 8i 


0693 


0906 


I118 


1330 


! 542 


205 


3"754 


1966 


2177 


2389 


2600 


2812 


3° 2 3 


3234 


3445 


3656 


206 


3867 


4078 


4289 


4499 


4710 


4920 


5i3 


534° 


555i 


5760 


207 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


743 6 


7646 


7854 


208 


8063 


8272 


8481 


8689 


8898 


9106 


93M 


9522 


973° 


9938 


209 
210 


320146 


0354 

2426 


0562 
2633 


0769 
2839 


0977 


1184 


I39 1 

3458 


1598 
3665 


1805 


2012 


322219 


3046 


3252 


4077 


211 


4282 


4488 


4694 


4899 


5105 


53 10 


5516 


5721 


5926 


6131 
8176 


212 


633 6 


6541 


6745 


6950 


7155 


7359 


7563 


7767 


7972 


213 


8380 


8583 


8787 


8991 


9194 


9398 


9601 


9805 


°oo8 


°2II 


214 


33°4i4 


0617 


0819 


1022 


1225 


1427 


1630 


1832 


2034 


2236 


215 


2438 


2640 


2842 


3°44 


3246 


3447 


3 6 49 


3850 


4051 


4253 


216 


4454 


4655 


4856 


5057 


5257 


5458 


5<>5« 


58S9 


6059 


6260 


217 


6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 


218 


8456 


8656 


8855 


9054 


9253 


945 1 


9650 


9849 


°°47 


°2 4 6 


219 


340444 


0642 


0841 


IQ 39 
3 


1237 


1435 
5 


1632 


1830 

7 


2028 
8 


2225 


N. 





1 


2 


4 


6 


9 



23 



21 



= — _ — _ 

N. 220. ZiOGvAJtfTHMS. Log. 342. 




N. 





1 

2620 
4589 

6549 
8500 

0442 

2 375 
4301 

6217 

8125 

O2 5 


2 


3 


4 


5 


6 


7 

3802 
5766 
7720 
9666 
1603 

353* 

5452 

7363 
9266 

»i6i 

3048 
4926 
6796 
8659 
V3 
2360 
4198 
6029 

7852 
9668 

1476 

3 2 77 
5070 
6856 
8634 

°4°5 
2169 
3926 
5676 
7419 

9*54 

0883 
2605 
4320 
6029 

773 1 
9426 
x ii4 
2796 
447^ 
6141 
7804 
9460 

2754 

4392 
6023 
7648 
9268 
°88i 


8 


9 




220 
221 
222 
223 
224 

| 225 

226 
227 
228 
229 

230 

231 
232 
233 
234 

235 

236 
237 
238 
239 

240 
241 
242 
243 
244 

245 
246 
247 
248 

249 


34 2 4 2 3 
4392 
6353 
8305 

350248 

2183 
4108 
6026 
7935 
9835 


2817 
4785 
6744 
8694 
0636 

2568 

4493 
6408 
8316 

°2I5 

2105 
3988 
5862 
7729 
9587 
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3280 

5"5 

6942 

8761 

0573 
2377 
4*74 
5964 
7746 

9520 

1288 
3048 
4802 
6548 


3014 
4981 

6939 
8889 
0829 

2761 
4685 

6599 
8506 

°4°4 
2294 
4176 
6049 

79*5 
9772 

1622 

3464 
5298 
7124 

8943 

0754 
2557 

4353 
6142 

7923 

9698 
1464 
3224 

4977 
6722 

8461 

°I92 

1917 

3635 
5346 
7051 
8749 
0440 
2124 
3803 

5474 
7139 
8798 
0451 
2097 

3737 
537i 

6999 
8621 
0236 

"1846 

345° 
5048 
6640 
8226 

9806 
1381 
2950 

4513 
6071 


3212 
5178 

7135. 
9083 
1023 

2954 
4876 
6790 
8696 
0593 


3409 

5374 
7330 
9278 
1216 

3 X 47 

5068 
6981 
8886 
0783 


3606 
557o 
7525 
9472 
1410 

3339 

5260 
7172 
9076 
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2859 

4739 
6610 

8473 
0328 

2175 
4015 

5846 
7670 
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1296 

3°97 
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375 1 
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7245 
8981 
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2433 
4149 
5858 

7561 
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0946 
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A^A 

5974 
7638 
9295 

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2590 

4228 
5860 
7486 
9106 
O720 

2328 

393° 
5526 
7116 
8701 

°279 
1852 

3419 
4981 

6537 


3999 
5962 

79*5 
9860 
1796 

3724 
5643 
7554 
9456 
J 35° 
3236 

5"3 

6983 
8845 
0698 

2544 
4382 
6212 
8034 
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5249 
7034 
8811 

0582 

2345 
4101 

5850 
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9328 
1056 
2777 

4492 
6199 

7901 

9595 
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2964 

4639 

6308 

7970 

9625 

l *7S 
2918 

4555 
6186 

7811 
9429 

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2649 

4249 
5844 

7433 
9017 

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2166 

373 2 
5293 
6848 

8 


4196 
6157 
8110 
0054 
1989 

3916 

5834 

7744 
9646 

J 539 




361728 
3612 
5488 

7356 
9216 

371068 
2912 
4748 
6577 
8398 

380211 
2017 

3815 

5606 

739° 
9166 

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2697 
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1917 

3800 

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7542 
9401 

1253 

3096 

4932 

6759 
8580 

0392 

2197 

3995 
5785 
7568 

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2873 

4627 

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8114 

9847 

1573 
3292 

5°°5 
6710 
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3467 
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949 J 

1066 

2637 

4201 

576o 

1 


2482 

4363 
6236 

8101 

9958 

1806 

3647 
5481 
7306 
9124 


2671 
455i 
6423 
8287 

°i43 
1991 

3831 
5664 
7488 
9306 

1115 

2917 
4712 

6499 
8279 

°o5i 
1817 

3575 
5326 

7071 

8808 
0538 
2261 

3978 
5688 

739 1 

9087 

°777 
2461 

4137 
5808 
7472 
9129 
O781 
2426 

4065 
5697 
7324 
8944 
°559 


3424 

53 QI 
7169 
9030 
0883 

2728 

4565 
6394 
8216 
o3o 

1837 
3636 

5428 
7212 
8989 

°759 
2521 

4277 
6025 
7766 




0934 
2737 

4533 
6321 
8101 

9875 
1641 
3400 
5152 
6896 




250 
251 
252 
253 
254 

255 

256 

, 257 

[ 258 

259 


39794° 
9674 

401401 
3121 
4 8 34 
6540 
8240 

9933 

411620 
3300 


8287 

O2O 
1745 
3464 
5176 
688l 

8579 
O27I 
1956 
3635 


8634 
o 3 6 5 
2089 
3807 
5517 
7221 
8918 
°6o9 
22q3 
397° 


9501 

J 228 
2949 
4663 
637O 

807O 

9764 
I45I 

3 J 3 2 

4806 




260 
261 
262 
263 
264 

265 
266 

| 267 
268 
269 


4*4973 
6641 
8301 
9956 

421604 

3246 
4882 
6511 

8i35 
9752 

431364 
2969 

4569 
6163 

775i 

9333 

440909 
2480 
4045 
5604 


5307 
6973 
8633 
O286 

1933 

3574 
5208 
6836 
8459 
°°75 
1685 
3290 
4888 
6481 
8067 

9648 

1224 
2793 
4357 
59 J 5 


5641 
7306 
8964 
°6i6 
2261 

3901 

5534 
7161 

8783 
0398 


6474 
8i35 
9791 

M-39 
3082 

4718 

6349 
7973 

959 1 
1203 




270 
271 
272 
273 
274 

275 
276 

277 
278 
279 


2007 
3610 
5207 
6799 
8384 

9964 
1538 
3106 
4669 
6226 


2167 
377° 
5367 
6957 
8542 

°I22 
1695 

3263 
4825 
6382 

5 


2488 
4090 
5685 

7275 
8859 

°437 
2009 

3576 

5137 
6692 


2809 
4409 
6004 
759 2 
9*75 

0752 
2323 
3889 

5449 
7003 







2 


3 


4 


6 


7 


9 





22 



— — 1 

N. 280. LOGARITHMS. Log. 447. 


N. 





1 


2 


3 


4 


5 


6 

8088 

9633 
1172 

2706 

4235 

5758 
7276 
8789 
0296 
1799 


7 
8242 

9787 
1326 
2859 
4387 
59io 
7428 
8940 

°447 
1948 

3445 
4936 
6423 
7904 
9380 

0851 
2318 

3779 
5235 
6687 


8 


9 


280 
281 
282 
283 
284 

285 

286 
287 
288 
289 


447158 
8706 

450249 
1786 
33i8 
4845 
6366 
7882 

939 2 
460898 


7313 
8861 
0403 
1940 
347i 

4997 
6518 
8033 

9543 

1048 

2548 
4042 

553 2 
7016 

8495 
9969 
1438 
2903 
4362 
5816 

7266 
8711 
0151 
1586 
3016 

4442 

5863 
7280 
8692 
°o99 

1502 

2900 
4294 
5683 
7068 

8448 
9824 
1196 
2564 
3927 

5286 
6640 
7991 

9337 
0679 

2017 

335i 
4681 
6006 
7328 

8646 

9959 
1269 

2575 
3876 

5174 
6469 

7759 
9045 
0328 


7468 
9015 

0557 
2093 
3624 

5i5o 
6670 
8184 
9694 
1198 


7623 
9170 
07 1 1 
2247 
3777 
5302 
6821 
8336 
9845 
_i348 
2847 
4340 
5829 
7312 
8790 

0263 
1732 

3*95 
4653 
6107 

7555 

8999 
0438 
1872 
33° 2 
4727 
6147 
7563 

8974 
0380 


7778 
9324 
0865 
2400 
3930 

5454 
6973 
8487 

9995 
1499 


7933 
9478 
1018 

2553 
4082 

5606 
7125 
8638 
°i 4 6 
1649 


8397 
9941 

1479 
3012 
4540 

6062 

7579 
9091 

°597 

2098 

3594 
5085 
6571 
8052 
9527 
o 99 8 
2464 

3925 
538i 
6832 


8552 ; 

°°95 j 
1633 j 

3 l6 5 1 
4692 | 

6214 

773 1 
9242 
C748 
2248 

3744 
5234 
6719 

82OO ! 
9675 

1145 
26lO 
407I 
5526 
6976 


290 
291 
292 
293 
294 

295 

296 
297 
298 
299 


462398 
3S93 

5383 
6868 

8347 
9822 
471292 
2756 
4216 
5671 


2697 

4191 

5680 
7164 
8643 

°n6 

1585 

3°49 
4508 
5962 

741 1 
8855 
0294 
1729 
3*59 

4585 
6005 
7421 
8833 
°239 
1642 
3040 

4433 
5822 

7206 

8586 
9962 

1333 

2700 

4 o6 3 
5421 
6776 
8126 

947i 
0813 

2151 

3484 

4813 
6139 
7460 


2997 
4490 

5977 
7460 

8938 

°4io 

1878 

334i 

4799 
6252 


3146 

4639 
6126 
7608 
9085 

°557 
2025 

3487 
4944 
6397 


3296 
4788 
6274 

7756 
9233 

°7°4 
2171 

3633 
5090 
6542 

7989 
9431 
0869 
2302 
373° 

5*53 

6572 

7986 

9396 
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2201 

3597 
4989 
6376 
7759 
9137 
°5ii 
1880 
3246 
4 6o 7 
5964 
7316 
8664 
°oo9 
1349 
2684 
4016 

5344 
6668 
7987 


300 
301 
302 
303 
304 

305 

306 
307 
308 
309 


477121 
8566 

480007 
1443 
2874 

4300 

572i 
7138 
8551 

995 8 

491362 

2760 

4 J 55 

5544 
6930 

8311 
9687 
501059 
2427 
379 1 


7700 
9143 
0582 
2016 

3445 
4869 
6289 
7704 
9114 

52O 


7844 
9287 
0725 
2159 
3587 
5011 
6430 
7845 
9255 
°66i 

2062 

3458 
4850 
6238 
7621 

8999 
°374 
1744 
3109 

447 ! 


8i33 
9575 
1012 

2445 
3872 

5295 
6714 
8127 
9537 
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2341 
3737 
5128 
6515 
7897 

9275 
0648 
2017 
3382 
4743 
6099 

745i 
8799 
0143 
1482 

2818 
4149 
5476 
6800 
8119 

9434 
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2053 

335 6 
4656 

595i 

7243 
8531 

9815 

1096 

7 


8278 
9719 
1156 
2588 
4015 

5437 
6855 

8269 

9677 
^81 

2481 

3876 
5267 
6653 

8035 
9412 
0785 
2154 
3518 
4878 
6234 
7586 

8934 
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1616 

2951 

4282 
5609 
6932 
8251 

9566 
°8 7 6 
2183 
3486 
4785 
6081 
7372 
8660 

9943 
1223 

8 


8422 
9863 
I299 
273I 
4157 

5579 
6997 

8410 
9818 

! 222 


310 
311 
312 
313 
314 

315 
316 
317 
318 
319 

~32F 
321 
322 
323 
324 

325 
326 
327 

328 
329 


1782 

3*79 
4572 
5960 

7344 
8724 
°o99 
1470 
2837 
4199 

5557 
6911 
8260 
9606 
0947 

2284 
3617 

4946 
6271 

7592 

8909 

°22I 

153° 

2835 
4136 

5434 
6727 

8016 
9302 

0584 


1922 

33 J 9 

4711 
6099 

7483 
8862 
0236 
1607 

2973 
4335 


2621 

40I5 
5406 
679I 

8173 

955° 

°922 
229I 

3655 

5 OI 4 
6370 
7721 
9068 
°4ii 
1750 

3084 
4415 
5-4i 
7064 
8382 


5°5i5° 
6505 

7856 

9203 

5 IQ 545 

1883 

3218 

4548 

5874 
7196 


5693 
7046 

8395 
9740 
1081 

2418 

375° 
5079 
6403 

7724 


5828 
7181 

8530 
9874 

1215 

2551 

3883 
5211 

6535 
7855 


330 
331 
332 
333 
334 

335 

336 
337 
338 
339 


518514 
9828 

521138 
2444 
3746 

5°45 
6339 
7630 
8917 
530200 


8777 
o9o 
1400 
2705 
4006 

53°4 
6598 
7888 

9174 
0456 


9040 

°353 
1661 
2966 
4266 

5563 
6856 

8145 
9430 
0712 


9171 
0484 
1792 
3096 
4396 

5 6 93 

6985 

8274 

9559 
0840 

5 


93°3 
°6i5 
1922 
3226 
4526 

5822 
7114 

8402 
9687 
0968 


9697 
•007 

23H 
3616 

49*5 
6210 
7501 
8788 
O072 
1351 

9 


N. 


1 


2 


3 


4: 


6 



23 



N. 340. XiOG-ARXTHXMES. Log. 531. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


340 
341 
342 
343 
344 

345 

346 
347 
348 
349 

351 
352 
353 
354 

355 
356 
357 
358 
359 


53*479 
2754 
4026 
5294 
6558 

7819 

9076 
540329 

1579 
2825 


1607 
2882 

4*53 
5421 
6685 

7945 
9202 

°455 

1704 

J2950 

4192 

543i 
6666 

7898 
9126 

0351 

1572 
2790 
4004 
5215 

6423 

7627 
8829 

O26 
1221 

2412 

3600 

4784 
5966 

7H4 
8319 
949I 
0660 
1825 
2988 

4H7 

53°3 
6457 

7607 

8754 
9898 
1039 
2177 
3312 
4444 

5574 
6700 
7823 

8944 
0061 

1176 

2288 

3397 

45°3 
5606 

6707 
7805 
8900 
9992 
1082 


1734 
3009 
4280 

5547 
6811 

8071 
9327 
0580 
1829 
3°74 
4316 

5555 
6789 
8021 
9249 

0473 
1694 
2911 
4126 

533_ 6 
6544 
7748 
8948 
°i 4 6 
1340 

253 1 
37i8 

49°3 
6084 
7262 

8436 
9608 
0776 
1942 
3104 

4263 

5419 
6572 

7722 

8868 

°OI2 

"53 
2291 
3426 
4557 
5686 
6812 

7935 
9056 
o I?3 

1287 
2399 
3508 
4614 
5717 
6817 

79H 
9009 

°IOI 

1191 


1862 

3i3 6 

4407 
5674 
6 937 
8197 
9452 
0705 

1953 
3 J 99 

4440 
5678 
6913 
8144 
9371 

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1816 

3°33 

4 2 47 
5457 
6664 
7868 
9068 
0265 

1459 

2650 

3837 
5021 
6202 
7379 


1990 
3264 

4534 
5800 
7063 

8322 

9578 
0830 
2078 
33 2 3 
4564 
5802 
7036 
8267 
9494 
0717 
1938 

3*55 
4368 

5578 


2117 

339i 
4661 
5927 
7189 

8448 
9703 

°955 
2203 

3447 
4688 
5925 
7i59 
8389 
9616 

0840 
2060 
3276 

4489 
5699 

6005 
8108 
9308 
0504 
1698 

2887 
4074 
5257 
6437 
7614 

8788 

9959 
1126 
2291 
3452 
4610 

5765 
6917 
8066 
9212 


2245 

35i8 

4787 
6053 

73*5 

8574 
9829 
1080 
2327 
'3571 
4812 
6049 
7282 
8512 
9739 
0962 
2181 

3398 
4610 
5820 

7026 
8228 
9428 

°62 4 

1817 

3006 
4192 
5376 
6 555 
773* 


2372 
3645 
4914 
6180 
7441 

8699 

9954 
1205 
2452 
3 6 9 6 
j.936 
6172 

74°5 
8635 
9861 

1084 
2303 
3519 
473 1 
_S94_o 
7146 

8349 
9548 
0743 
1936 

3 I2 5 

4311 

5494 
6673 

7849 


2500 

377* 
5041 
6306 
7567 

8825 
O o79 
1330 
2576 
3820 

5060 
6296 
7529 
8758 
9984 

1206 
2425 
3640 
4852 
6061 

7267 
8469 
9667 
0863 
2055 

3^44 
4429 
5612 
6791 
79 6 7 
9140 
O309 
1476 
2639 
3800 

4957 
611 1 

7262 
8410 

9555 
0697 
1836 
2972 
4105 
5235 
6362 
7486 
8608 
9726 

°8 4 2 

1955 

3064 

4171 
5276 
6377 
7476 
8572 
9665 

°755 
1843 


2627 
3899 
5167 
6432 
7693 

8951 

2O4 

1454 

2701 
3944 


544068 
53°7 
6543 
7775 
9003 

550228 
1450 
2668 

3883 
5094 

556303 
7507 
8709 
9907 

561101 

2293 
3481 
4666 
5848 
7026 


5183 
6419 
7652 
8881 
°io6 

1328 

2547 
3762 

4973 
6182 


360 
361 
362 
363 
364 

365 

366 
367 
368 
369 


6785 
7988 
9188 
0385 
1578 

2769 
3955 
5i39 
6320 

7497 


7387 
8589 
9787 
0982 
2174 

33 62 
4548 

573° 
6909 
8084 


370 
371 
372 
373 
374 

375 
376 
377 
378 
379 

"380 
381 
382 
383 

384 

385 

386 
387 
388 
389 

390 

391 
392 
393 
394 

1 395 
396 
397 
398 
399 




568202 
9374 

57°543 
1709 
2872 

4031 
5188 
6341 

7492 

8639 

579784 

580925 

2063 

3*99 
43 3 L 

5461 
6587 
7711 
8832 

_995° 
591065 

2177 
3286 

4393 
549 6 
6597 
7695 
8791 

9883 
600973 


8554 
9725 
0893 
2058 
3220 

4379 

5534 
6687 
7836 
8983 

°I26 

1267 
2404 

3539 
4670 

5799 

6925 

8047 
91.67 
0284 

1399 
2510 
3618 
4724 
5827 
6927 
8024 
9119 

°2IO 
I299 


86 7I 
9842 

IOIO 

2174 
333 6 

4494 
5650 
6802 

795i 
9097 


8905 
0076 
1243 
2407 
3568 

4726 
5880 
7032 
8181 
9326 

~°469 
1608 

2745 

3879 
5009 

6137 
7262 
8384 

9503 
°6i9 

I73 2 
2843 

395° 
5055 
6i57 
7256 

8353 
9446 

°537 
1625 

6 


9023 
°i93 
1359 

2523 
3684 

4841 
5996 

7H7 
8295 
9441 

0583 
1722 
2858 
3992 
5122 

6250 

7374 
8496 
9615 
O730 

1843 
2954 
4061 
5165 
6267 

7366 
8462 

955 6 
°6 4 6 

'734 

7 


9257 
0426 
1592 
2755 
3 OI 5 
5072 
6226 

7377 
8525 
9669 


0241 
1381 
2518 
3652 
4783 
5912 
7037 
8160 
9279 
0396 


°355 
1495 
2631 

37 6 5 
4896 

6024 
7149 
8272 

939 1 

°5°7 
1621 
2732 
3840 

4945 
6047 

7146 
8243 
9337 

° 4 28 

1517 


°8n 
1950 

3085 
4218 

5348 

6475 
7599 
8720 

9838 
°953 
2066 

3*75 
4282 
5386 
6487 

7586 
8681 

9774 
0864 
1951 


1510 
2621 
3729 
4834 
5937 
7037 
8134 
9228 
o 3 i 9 
1408 





1 


2 


3 


4 


5 


8 


9 



24 



N. 


400. 




LOGARITHMS. 




Log. 602. 


N. 





1 


2 


3 

2386 


4 


5 


6 


7 


8 


9 


400 


602060 


2169 


2277 


2494 


2603 


2711 


2819 


2928 


3036 


401 


3H4 


3 2 53 


3361 


3469 


3577 


3686 


3794 


3902 


4010 


4118 


402 


4226 


4334 


4442 


455° 


4658 


4766 


4*74 


4982 


5089 


5*97 


403 


53°5 


5413 


5521 


5628 


5736 


5*44 


595i 


6059 


6166 


6274 


404 


6381 


6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


734* 


405 


7455 


7562 


7669 


7777 


7**4 


7991 


8098 


8205 


8312 


8419 


406 


8526 


8633 


8740 


8847 


8954 


9061 


9167 


9274 


93*i 


9488 


407 


9594 


9701 


9808 


9914 


O2I 


0128 


°234 


u 34i 


°447 


°554 


403 


610660 


0767 


0873 


0979 


I086 


1192 


1298 


1405 


1511 


1617 


409 


1723 


1829 


1936 


2042 
3102 


2148 


2254 


2360 


2466 
3525 


2572 


2678 


410 


612784 


2890 


2996 


3207 


33*3 


3419 


3630 


3736 


411 


3842 


3947 


4053 


4159 


4264 


4370 


4475 


45*i 


4686 


4792 


412 


4897 


5003 


5108 


5213 


53 J 9 


5424 


5529 


5 6 34 


574° 


5*45 


413 


595o 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6*95 


414 


7000 


7105 


7210 


73*5 


7420 


7525 


7629 


7734 


7*39 


7943 


415 


8048 


8i53 


8257 


8362 


8466 


8571 


8676 


8780 


8884 


8989 


416 


9093 


9198 


9302 


9406 


9511 


9615 


9719 


9824 


9928 


O032 


417 


620136 


0240 


0344 


0448 


0552 


0656 


0760 


0864 


0968 


1072 


418 


1176 


1280 


1384 


1488 


1592 


1695 


1799 


1903 


2007 


2110 


419 


2214 
623249 


2318 
3353 


2421 
3456 


2525 
3559 


2628 


2732 


2*35 
3869 


2939 
3973 


3042 


3146 


420 


3663 


3766 


4076 


4179 


421 


4282 


43*5 


4488 


459i 


4695 


479* 


4901 


5004 


5107 


5210 


422 


53* 2 


5415 


5518 


5621 


5724 


5*27 


5929 


6032 


6135 


6238 


423 


6340 


6443 


6546 


6648 


6751 


6853 


6956 


7058 


7161 


7263 


424 


7366 


7468 


7571 


7673 


7775 


7*7* 


7980 


8082 


8185 


8287 


425 


8389 


8491 


8593 


8695 


8797 


8900 


9002 


9104 


9206 


9308 


426 


9410 


9512 


9613 


9715 


9*i7 


9919 


O2I 


°I23 


°224 


0326 


427 


630428 


0530 


0631 


0733 


0835 


0936 


IO38 


"39 


I24I 


1342 


428 


1444 


1545 


1647 


1748 


i*49 


1951 


2052 


2153 


2255 


2356 


429 
430 


2457 
633468 


2559 
35 6 9 


2660 


2761 
377i 


2862 


2963 
3973 


3064 


3 l6 5 

4i75 


3266 
4276 


3367 


3670 


3872 


4074 


4376 


431 


4477 


457* 


4679 


4779 


4880 


4981 


5081 


5182 


5283 


53*3 


432 


5484 


55*4 


5685 


57*5 


5886 


59*6 


6087 


6187 


6287 


6388 


433 


6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


729O 


739o 


434 


7490 


7590 


7690 


7790 


7890 


7990 


809O 


8190 


829O 


8389 


435 


8489 


8589 


8689 


8789 


8888 


8988 


9088 


9188 


92*7 


93*7 


436 


9486 


9586 


9686 


97*5 


9**5 


9984 


0084 


°i8 3 


"283 


0382 


437 


640481 


0581 


0680 


0779 


0879 


0978 


1077 


1177 


I276 


1375 


438 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


439 
~44(T 


2465 


2563 
355i 


2662 
3650 


2761 
3749 


2860 


2959 


3058 


3 J 5 6 


3255 


3354 


6 43453 


3*47 


3946 


4044 


4H3 


4242 


434o 


441 


4439 


4537 


4636 


4734 


4832 


493i 


5029 


5127 


5226 


5324 


442 


5422 


55^1 


5619 


5717 


5815 


5913 


6011 


6110 


6208 


6306 


443 


6404 


6502 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7285 


444 


73 8 3 


7481 


7579 


7676 


7774 


7872 


7969 


8067 


8165 


8262 


445 


8360 


8458 


8555 


8653 


8750 


8848 


8945 


9043 


9I4O 


9237 


446 


9335 


9432 


9530 


9627 


9724 


9821 


9919 


°oi6 


°ii3 


°2IO 


447 


650308 


0405 


0502 


0599 


0696 


0793 


0890 


0987 


1084 


Il8l 


448 


1278 


1375 


1472 


1569 


1666 


1762 


i*59 


1956 


2053 


2150 


449 


2246 


2343 
33°9 


2440 
3405 


2536 
3502 


2633 


2730 


2826 


2923 


3 OI 9 
39*4 


3Il6 


450 


653213 


359* 


3695 


379i 


3888 


4080 


451 


4*77 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 


5042 


452 


5138 


5235 


533 1 


5427 


5523 


5619 


5715 


5810 


5906 


6002 


453 


6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6864 


6960 


454 


7056 


7152 


7247 


7343 


743* 


7534 


7629 


7725 


7820 


7916 


455 


8011 


8107 


8202 


8298 


8393 


8488 


8584 


8679 


*774 


887O 


456 


8965 


9060 


9i55 


9250 


9346 


944 * 


9536 


9631 


9726 


982I 


457 


9916 


°OII 


°io6 


2OI 


0296 


0391 


0486 


0581 


0676 


o 7?I 


458 


660865 


0960 


1055 


II50 


1245 


1339 


1434 


1529 


1623 


1718 


459 


1813 


1907 


2002 


2096 


2191 


2286 


2380 
6 


2475 
7 


2569 


2663 
9 


N. 





1 


2 


3 


4 


5 


8 



25 



N. 460. XiOa-ARITHRaS. Log. 662. 


N. 
460 
461 
462 
463 
464 

465 

466 
467 
468 
469 





1 

2852 
3795 
473 6 
5675 
6612 

7546 
8479 
9410 

°339 
1265 

2190 

3"3 

4°34 
4953 
5870 

6785 
7698 
8609 

95 T 9 

0426 

I33 2 
2235 

3*37 
4<>37 
4935 

5831 
6726 
7618 
8509 
9398 
0285 
1 1 70 
2053 

2 935 
3815 

4693 
5569 
6444 
7317 
8188 


2 


3 

3041 

3983 
4924 
5862 
6799 

7733 
8665 

9596 

0524 

i45i 

2375 
3 2 97 
4218 

5137 
6053 

6968 
7881 
8791 
9700 
0607 

15*3 

2416 

3317 
4217 

5"4 
6010 
6904 
7796 
8687 
9575 
0462 

1347 
2230 
3111 
399 1 
4868 

5744 
6618 

749 1 
8362 

9231 

0098 
0963 
1827 
2689 

3549 

4408 
5265 
6120 
6974 

7826 
8676 
9524 
0371 
1217 

2060 
2902 

3742 
4581 
5418 


4 


5 


6 

3324 
4266 
5206 
6143 
7079 

8013 
8945 

9875 
0802 
1728 

2652 
3574 
4494 
5412 
6328 

7242 
8154 
9064 

9973 
0879 

1784 
2686 

3587 
4486 

5383 
6279 
7172 
8064 
8953 
9841 
0728 
1612 
2494 
3375 
4254 

5131 

6007 
6880 

7752 
8622 

949 1 
°358 
1222 
2086 
2947 

3807 
4665 
5522 
6376 
7229 

"8081 
8931 
9779 
0625 
1470 

2313 

3154 
3994 
4833 
5 66 9 
6 


7 


8 

3512 

4454 

5393 
6331 
7266 

8199 
9131 
°o6o 
0988 
I9 J 3 
2836 

3758 
4677 

5595 
6511 

7424 
8336 
9246 

°i54 
1060 

1964 
2867 

37 6 7 
4666 

5563 
6458 

735i 
8242 

9i3* 
°oi9 

0905 
1789 
2671 

355i 

4430 

5307 
6182 

7°55 
7926 

8796 

9664 

°53i 
1395 
2258 

3"9 

3979 
4837 
5693 
6547 
7400 

8251 
9100 
9948 
0794 
1639 

2481 

3323 
4162 
5000 
5836 
8 


9 


662758 
3701 
4642 
5581 
6518 

7453 
8386 

9317 
670246 

"73 

672098 
3021 

3942 
4861 

5778 
6694 
7607 
8518 
9428 
680336 

681241 
2145 

3°47 
3947 
4845 

5742 
6636 
7529 
8420 
93°9 
690196 
1081 
1965 
2847 
37 2 7 
4605 
5482 
6356 
7229 
8101 


2947 
3889 
4830 
5769 
6705 

7640 
8572 

95°3 
0431 
1358 

2283 
3205 
4126 

5045 
5962 

6876 
7789 
8700 
9610 
0517 

1422 
2326 
3227 
4127 
5° 2 5 
5921 
6815 
7707 

8598 
9486 

0373 
1258 
2142 
3023 

39°3 
478i 
5 6 57 

6531 

7404 
8275 

9144 

°OII 

0877 
1741 
2603 

34 6 3 

4322 

5179 
6035 

6888 

7740 
8591 
9440 
0287 
1132 

1976 
2818 
3 6 59 
4497 
5335 


3 I 35 

4078 
5018 

595 6 
6892 

7826 

8759 
9689 
0617 

1543 


3 2 3° 
4172 
5112 
6050 
6986 

7920 
8852 
9782 
0710 
1636 


3418 
4360 

5299 
6237 

7*73 
8106 
9038 
9967 
0895 
1821 

2744 
3666 
4586 

55°3 
6419 

7333 
8245 

9155 
0063 
0970 

~i874 

2777 
3 6 77 
4576 
5473 
6368 
7261 

8i53 
9042 

993° 
0816 
1700 
2583 
34 6 3 
4342 
5219 
6094 
6968 

7839 
8709 

9578 

°444 
1309 
2172 
3°33 

3893 
475i 
5607 
6462 
73 I 5 
8166 
9015 
9863 
0710 

1554 
2397 
3238 
4078 
4916 

5753 
7 


3607 
4548 

5487 
6424 
7360 

8293 
9224 

°i53 
1080 
2005 


470 
471 
472 
473 
474 

475 
476 

477 
478 
479 


2467 
3390 
4310 
5228 
6i45 
7059 
7972 
8882 
9791 
0698 


2560 
3482 
4402 
5320 
6236 

7151 
8063 

8973 
9882 
0789 


2929 
3850 
4769 
5687 
6602 

7516 
8427 
9337 
0245 
1151 


480 
481 
482 
483 
484 

485 
486 
487 
488 
489 


1603 
2506 
34°7 
43°7 
5204 

6100 
6994 

7886 
8776 
9664 


1693 
2596 

3497 
4396 
5294 
6189 
7083 

7975 
8865 

9753 


2055 
2957 
3857 
4756 
5652 

6547 
7440 

8331 
9220 
io7 

0993 

1877 

2759 
3639 

45 J 7 

5394 
6269 

7142 
8014 
8883 

975i 

°6i7 
1482 

2344 
3205 

4065 
4922 

5778 
6632 

7485 
8336 
9185 

°°33 
0879 
1723 

2566 

34°7 
4246 
5084 
5920 


490 
491 
492 
493 
494 

495 

496 
497 
498 
499 


0550 

1435 
2318 

3199 

4078 

4956 

583 2 
6706 

7578 
8449 


0639 

1524 
2406 

3287 
4166 

5°44 
59*9 
6793 
7665 

8535 


500 
501 
502 
503 
504 

505 

506 
507 
508 
509 


698970 
9838 

700704 
1568 
2431 

3291 

4151 
5008 
5864 
6718 


9057 
9924 
0790 

1654 
2517 

3377 
4236 

5°94 
5949 
6803 

7655 
8506 

9355 
0202 
1048 

1892 

2734 
3575 
4414 
5251 

1 


9317 
°i8 4 
1050 
1913 

2775 

3 6 35 
4494 
535° 
6206 
7059 


9404 
O271 
1136 
1999 
2861 

3721 
4579 
543 6 
6291 
7144 


510 
511 
512 
513 
514 

515 

516 
517 

518 
519 


707570 
8421 
9270 

710117 
0963 

1807 
2650 

349 1 
4330 

5167 


7911 
8761 
9609 
0456 
1301 

2144 
2986 
3826 
4665 
550 2 


7996 
8846 
9694 
0540 
1385 

2229 
3070 
3910 

4749 
J586 

5 


N. 





2 


3 


4 


9 



26 



N. . 


520. 


LOGARITHMS. 




Log. 


— ^ 

716. ; 


N. 1 





1 


2 


3 

6254 


4 


5 


6 


7 
6588 


8 

6671 


9 1 


520 


716003 


6087 


6170 


6337 


6421 


6504 


6 754 
7587 


521 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


522 


7671 


7754 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 


523 


8502 


8585 


8668 


8751 


8834 


8917 


9000 


9083 


9165 


9248 1 


524 


933 1 


9414 


9497 


9580 


9663 


9745 


9828 


9911 


9994 


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525 


720159 


0242 


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0490 


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0655 


0738 


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0903 


526 


0986 


1068 


1151 


1233 


1316 


1398 


1481 


1563 


1646 


1728 


527 


1811 


1893 


!975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 


528 


2634 


2716 


2798 


2881 


2,963 


3°45 


3127 


3209 


3291 


3374 


529 


345 6 


3538 


3620 


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4522 


3784 


3866 
4685 


3948 


4030 


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4194 

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530 


724276 


4358 


4440 


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4849 


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5°95 


5176 


5258 


534° 


5422 


5503 


5585 


5667 


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5830 


532 


59 12 


5993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 


533 


6727 


6809 


6890 


6972 


7053 


7 X 34 


7216 


7297 


7379 


7460 


534 


754 1 


7623 


7704 


7785 


7866 


7948 


8029 


8110 


8191 


> 8273 


535 


8354 


8435 


8516 


8597 


8678 


87<59 


8841 


8922 


9003 


9084 


536 


9165 


9246 


9327 


9408 


9489 


9570 


9651 


9732 


9813 


9893 


537 


9974 


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0298 


°378 


°459 


0540 


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O702 


538 


730782 


0863 


0944 


1024 


1105 


1186 


1266 


1347 


1428 


1508 


539 
540 


1589 
73 2 394 


1669 
2474 


1750 


1830 
2635 


1911 


1991 


2072 
2876 


2152 
2956 


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2313 


2555 


2715 


2796 


3 IJ 7 


541 


3*97 


3278 


3358 


3438 


35i8 


3598 


3679 


3759 


3839 


3919 


542 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 


543 


4800 


4880 


4960 


5040 


5120 


5200 


5279 


5359 


5439 


55*9 


544 


5599 


5679 


5759 


5838 


59i8 


5998 


6078 


6157 


6237 


6317 


545 


6397 


6476 


6556 


6635 


6715 


6795 


6874 


6954 


7034 


7113 


546 


7193 


7272 


7352 


743 1 


75 11 


7590 


7670 


7749 


7829 


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547 


7987 


8067 


8146 


8225 


8305 


8384 


8463 


8543 


8622 


8701 


548 


8781 


8860 


8939 


9018 


9097 


9177 


9256 


9335 


9414 


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549 


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973i 


9810 


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9968 

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550 


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0521 


0600 


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1073 


551 


1152 


1236 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


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552 


1939 


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2096 


2175 


2254 


2332 


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2489 


2568 


2647 


553 


2725 


2804 


2882 


2961 


3°39 


3118 


3196 


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3353 


343i 


554 


3510 


3588 


3667 


3745 


3823 


3902 


3980 


4058 


4136 


4215 


555 


4293 


437i 


4449 


4528 


4606 


4684 


4762 


4840 


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556 


5°75 


5*53 


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53°9 


5387 


5465 


5543 


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5 6 99 


5777 


557 


5855 


5933 


601 1 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


558 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 


7256 


7334 


559 
560 


7412 


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7567 
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7645 


7722 


7800 


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9 J 95 


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563 


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1048 


1125 


1202 


564 


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1433 


1510 


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1818 


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1972 


565 


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2125 


2202 


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2509 


2586 


2663 


2740 


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2893 


2970 


3047 


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3200 


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3353 


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3506 


567 


3583 


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3813 


3889 


3966 


4042 


4119 


4195 


4272 


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434 8 


4425 


4501 


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473o 


4807 


4883 


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569 


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5189 
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6027 


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54i7 


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570 


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6256 


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571 


6636 


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6940 


7016 


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7244 


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7548 


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575 


9668 


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576 


760422 


0498 


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0724 


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0875 


0950 


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577 


1176 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 


578 


1928 


2003 


2078 


2153 


2228 


2303 


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2529 


2604 


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2829 


2904 
3 


2978 


3°53 
5 


3128 

6 


3203 
7 


3278 
8 


3353 

9 

_ 


N. 





2 


4: 



27 



1 N. 580. LOaARITHIVIS. Log. 763. 


N. 





1 


2 


3 


4 


5 


6 

3877 
4624 

537o 
6115 
6859 

7601 
8342 
9082 
9820 
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1293 
2028 
2762 

3494 
4225 

4955 
5683 
641 1 

7137 
7862 

"8585 
9308 

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0749 
1468 

2186 

2902 

3618 
4332 

5°45 

5757 
6467 
7177 
7885 
8593 
9299 
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0707 
1410 
2111 

2812 

35" 

4209 
4906 
5602 

6297 
6990 
7683 

8374 
9065 

9754 
0442 
1129 
1815 
2500 

3184 
3867 

4548 
5229 
5908 

6 


7 


8 


9 


580 
581 

582 
583 

5S4 

585 
586 
5 8 7 
588 

5S9 


763428 
4176 
4923 
5669 
6413 
7156 
7898 
8638 

9377 
770115 


35°3 
4251 
4998 

5743 
6487 

7230 
7972 
8712 

945i 
0189 

0926 
1661 
2395 
3128 
3860 

4590 

53*9 
6047 
6774 
7499 
8224 
8947 
9669 
0389 
1109 
1827 

2 544 
3260 

3975 
4689 

5401 
6112 
6822 

753 1 
8239 

8946 
9651 
0356 
1059 
1761 

2462 
3162 
3860 
4558 
5 2 54 

5949 

6644 

7337 
8029 
8720 


3578 
4326 
5072 
5818 
6562 

73°4 
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9525 
0263 

0999 

1734 
2468 
3201 
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4663 

539 2 
6120 
6846 
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3 6 53 

4400 

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6636 

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8860 
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1073 
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6193 
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8368 

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1253 

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3403 
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8934 
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3802 

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5296 
6041 

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3952 
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6 933 
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4027 

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5520 
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7749 
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1440 
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5100 
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7823 
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590 
591 
592 
593 
594 

595 
596 
597 

598 
599 

600 
601 
602 
603 
604 

605 

606 
607 
608 
609 


770352 

1587 
2322 

3°55 
3786 

45 J 7 
5 2 46 

5974 
6701 
7427 

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8874 
9596 

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1037 

1755 

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39°4 
4617 


1 146 

1881 
2615 
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4079 

4809 

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1220 

1955 
2688 

342i 
4152 

4882 
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7064 
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1367 
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2835 
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2114 
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6396 
7106 
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9228 

9933 

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1340 
2041 


610 
611 
612 
613 
614 

615 

616 
617 
618 
619 


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6041 
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8875 
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0988 
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6713 
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1541 
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6325 

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5970 
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620 
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622 
623 
624 

625 
626 
627 

628 
629 


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2602 
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3022 

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5811 

65 5 
7198 
7890 
8582 
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630 
631 
632 
633 
634 

635 
636 
637 
638 
039 


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800029 

0717 

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2774 
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55 6 9 

1 


9547 
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9961 
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2705 j 

3389 

407I 

4753 

5433 
6112 

9 


N. 





2 


3 


4 


5 



28 



K. 640. EOOA1LXTH1II& Log. 806. 


N.- 





1 


2 


3 

6384 
7061 
7738 
8414 
9088 

9762 

°434 
1106 
1776 
2445 

3"4 

378i 
4447 
5"3 
5777 
6440 
7102 
7764 
8424 
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974i 
0399 
1055 
1710 
2364 

3018 
3670 
4321 
4971 
5621 

6269 
6917 

75 6 3 
8209 

8853 

9497 
0139 

0781 

1422 

2062 

2700 

3338 

3975 
4611 

5M7 
5881 
6514 
7146 

7778 
8408 

9038 
9667 
0294 
0921 
1547 
2172 
2796 
3420 
4042 
4664 


4 


5 


6 


7 


8 


9 


640 
641 
642 
643 
644 

645 
646 
647 
648 
649 


806180 
6858 

7535 
8211 
8886 

9560 

810233 

0904 

1575 
2245 


6248 
6926 
7603 
8279 
8953 
9627 
0300 
0971 
1642 
2312 

2980 

3648 

43 J 4 
4980 
5644 

6308 
6970 
7631 
8292 
895i 
9610 
0267 
0924 

1579 

2233 

2887 

3539 
4191 

4841 
549i 
6140 
6787 

7434 
8080 
8724 

9368 

°OII 

0653 
1294 
1934 

2573 
3211 

3848 

4484 
5120 

5754 
6387 
7020 
7652 
8282 

8912 

954i 
0169 
0796 
1422 

2047 
2672 

3 2 95 
39i8 

4539 

1 


6316 
6994 
7670 
8346 
9021 

9694 
0367 
1039 
1709 
2379 

3°47 
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438i 
5046 

57" 

6374 
7036 
7698 

8358 
9017 

9676 
0333 
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1645 
2299 

2952 

3605 

4256 

4906 

5556 

6204 

6852 

7499 
8144 

8789 
9432 
0075 
0717 
1358 
1998 

2637 

3275 
3912 

4548 
5183 
5817 
6451 
7083 
77*5 
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8975 
9604 
0232 
0859 
1485 

2110 
2734 

3357 
3980 
4601 


6451 
7129 
7806 
8481 
9156 

9829 

0501 

"73 
1843 

2512 


6519 

7197 
7873 
8549 
9223 

9896 
0569 
1240 
1910 

2579 


6587 
7264 
7941 
8616 
9290 

9964 
0636 
1307 
1977 
2646 


6655 

7332 
8008 
8684 
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1374 
2044 
2713 

338i 

4048 
4714 

5378 
6042 

6705 

73 6 7 
8028 
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13*7 
1972 
2626 

3 2 79 
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5231 
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8030 
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1 1 72 
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2422 
3046 
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7 


6723 
7400 
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0098 

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2111 
2780 

3448 
4114 

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5445 
6109 

6771 

7433 
8094 

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9412 


6790 
7467 
8143 
8818 
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0165 
0837 
1508 
2178 
2847 


l 650 j 812913 

651 3581 

652 J 4248 

653 j 4913 

654 5578 

655 j 6241 

656 1 6904 

657 7565 

658 8226 

659 8885 


3181 
3848 

45 14 
5179 
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6506 
7169 
7830 
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3 2 47 
39 J 4 
4581 
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6 573 
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3314 

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53 12 
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7301 
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9939 
0595 
1251 

1906 
2560 

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4516 
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6464 
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1614 
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353° 
4166 
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5437 
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733 6 
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9227 

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1 109 

1735 
2360 

2983 
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4229 
4850 


35 J 4 
4181 
4847 

55" 

6175 

6838 

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8160 
8820 
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660 
661 
662 
663 
664 

665 

666 
667 
668 
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8i9544 

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0858 

1514 

2168 

2822 

3474 
4126 
4776 
5426 


9807 
0464 
1120 

1775 
2430 

3 o8 3 

3735 
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5 36 
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9873 
0530 
1186 
1841 
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3148 
3800 

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5101 

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6399 
7046 
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8982 

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0909 
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2037 
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399 6 
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1 1 02 
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3 6 57 
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6830 
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1448 j: 
2103 
2756 

34°9 
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6010 


670 
671 
672 
673 
674 

675 

676 
677 
678 
679 

680 

681 

682 
683 
684 

685 

686 
687 
688 
689 


826075 
6723 

73 6 9 
8015 
8660 

9304 

9947 

830589 

1230 

1870 

832509 

3 J 47 
3784 
4421 
5056 

5691 
6324 
6957 
7588 
82 1 9 

838849 

9478 
840106 

°733 
1359 
1985 
2609 
3 2 33 
3855 
4477 


6 334 
6981 
7628 

8273 
8918 

9561 

2O4 

0845 
i486 

2126 


6658 
73°5 
795i 
8595 
9^39 
9882 
0525 
1166 
1806 

2445 


2764 
3402 
4039 
4675 
5310 

5944 
6577 
7210 
7841 
8471 


2828 

3466 

4103 

4739 

5373 

6007 

6641 

7273 
7904 

8534 


3083 : 
3721 .; 
4357 
4993 
5627 

6261 
6894 

7525 
8156 
8786 


690 
691 
692 
693 
694 

695 

696 
697 
698 
699 


9101 

9729 
o357 
0984 
1610 

2235 
2859 
3482 
4104 
4726 


9164 
9792 
0420 
1046 
1672 

2297 
2921 

3544 
4166 

4788 
5 


9352 
9981 
0608 
1234 
i860 

2484 
3108 
373 1 
4353 

4974 


9415 

°°43 
0671 

1297 

1922 

2547 
3170 

3793 

44*5 

5036 


1 H. 





2 


3 


4 


6 


8 


9 



29 



N. 


700. LOGARITHMS. Log. 845. 


N. 





1 


2 


3 

5284 


4 


5 


6 


7 
553 2 


8 


9 


700 


845098 


5160 


5222 


5346 


5408 


547o 


5594 


5656 


701 


5718 


5780 


5842 


59°4 


5966 


6028 


6090 


6151 


6213 


6275 


702 


6337 


6 399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


703 


6 955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 


704 


7573 


7 6 34 


7696 


775^ 


7819 


7881 


7943 


8004 


8066 


8128 


705 


8189 


8251 


8312 


8374 


8435 


8497 


8«9 


8620 


8682 


8743 


706 


8805 


8866 


8928 


8989 


9051 


9112 


9 J 74 


9235 


9297 


9358 


707 


9419 


9481 


9542 


9604 


9665 


9726 


9788 


9849 


9911 


9972 


708 


850033 


0095 


0156 


0217 


0279 


0340 


0401 


0462 


0524 


0585 


709 
710 


0646 
851258 


0707 
1320 


0769 


0830 


0891 


°95 2 
1564 


1014 

1625 


1075 
1686 


1136 

1747 


1197 


1381 


1442 


I5°3 


1809 


711 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 


712 


2480 


2541 


2602 


2663 


.2724 


2785 


2846 


2907 


2968 


3029 


713 


3090 


3*5° 


3211 


3272 


3333 


3394 


3455 


35I 6 


3577 


3 6 37 


714 


3698 


3759. 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 


715 


4306 


43 6 7 


4428 


4488 


4549 


4610 


4670 


473 1 


4792 


'4852 


716 


4913 


4974 


5034 


5°95 


5156 


5216 


5277 


5337 


5398 


5459 


717 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 


718 


6124 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 


719 

~72T 


6729 
857332 


6789 
7393 


6850 


6910 


6970 


7031 
7634 


7091 


7'5^ 

7755 


7212 


7272 


7453 


75i3 


7574 


7694 


7815 


7875 


721 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 


722 


8537 


^597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


9078 


723 


9138 


9198 


9258 


9318 


9379 


9439 


9499 


9559 


9619 


9679 


724 
725 


9739 


9799 


9859 


9918 


9978 


0038 


0098 


0158 


°2l8 


0278 


860338 


0398 


0458 


0518 


0578 


0637 


0697 


0757 


0817 


0877 


726 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 


727 


1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 


728 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 


729 


2728 


2787 
3382 


2847 


2906 
35 QI 


2966 


J025 
3620 


3 o8 5 
3680 


3144 
3739 


3204 
3799 


3263 


730 


863323 


344 2 


356i 


3858 


731 


39*7 


3977 


4036 


4096 


4155 


4214 


4274 


4333 


4392 


445 2 


732 


4511 


457o 


4630 


4689 


4748 


4808 


4867 


4926 


4985 


5°45 


733 


5 io 4 


5 l6 3 


5222 


5282 


534i 


5400 


5459 


5519 


5578 


5 6 37 


734 


5696 


5755 


58H 


5^74 


5933 


5992 


6051 


6110 


6169 


6228 


735 


6287 


6346 


6405 


6465 


6524 


6583 


6642 


6701 


6760 


6819 


736 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


737 


7467 


7526 


75^5 


7644 


7703 


7762 


7821 


7880 


7939 


7998 


738 


8056 


8115 


8174 


8*33 


8292 


8350 


8409 


8468 


8527 


8586 


739 
740 


8644 
869232 


8703 
9290 


8762 
9349 


8821 
9408 


8879 


8938 


8997 
9584 


9°5 6 
9642 


9114 
9701 


9*73 
9760 


9466 


95*5 


741 


9818 


9877 


9935 


9994 


0053 


°iii 


O170 


°228 


0287 


°345 


742 


870404 


0462 


0521 


0579 


0638 


0696 


°755 


0813 


0872 


0930 


743 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


I398 


1456 


1515 


744 


1573 


163-1 


1690 


1748 


1806 


1865 


1923 


I981 


2040 


2098 


745 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 


746 


2739 


2797 


2*55 


2913 


2972 


3030 


3088 


3146 


3204 


3262 


747 


33 21 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844 


748 


3902 


3960 


4018 


4076 


4134 


4192 


4250 


4308 


4366 


4424 


749 
750 


4482 
875061 


4540 
5119 


4598 


4656 


47H 


4772 
535i 


4830 


4888 
5466 


4945 

55 2 4 


5003 
5582 


5*77 


5 2 35 


5 2 93 


54°9 


751 


5640 


5698 


575 6 


5813 


5871 


59 2 9 


5987 


6045 


6102 


6160 


752 


6218 


6276 


6 333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 


753 


6795 


6^53 


6910 


6968 


v 7026 


7083 


7141 


7199 


7256 


73H 


754 


737i 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 ! 


755 


7947 


8004 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 \ 


756 


8522 


8579 


8637 


8694 


8752 


8809 


8866 


8924 


8981 


9039 


. 757 


9096 


9*53 


9211 


9268 


93 2 5 


9383 


9440 


9497 


9555 


9612 


758 


9669 


9726 


9784 


9841 


9898 


9956 


°oi3 


"070 


°I27 


°i85 


759 


880242 


0299 
1 


°35 6 
2 


0413 


0471 


0528 


0585 


0642 
7 


0699 

IP 


0756 

9 

J 


N^ 





3 1 4 


5 


6 



30 



IT. 760. IiO&JZLTLlTHM&. Log. 880. 


N. 





1 

"0871 
1442 
2012 
2581 
3150 

3718 
4285 
4852 
5418 
5983 


2 


3 


4 


5 


6 

1156 

1727 
2297 
2866 
3434 
4002 
4569 

5*35 
5700 
6265 


7 


8 


9 


760 
761 
762 
763 
764 

765 
766 
767 
768 
769 


880814 

1385 
1955 
2525 

3°93 

3661 

4229 

4795 

53 61 
5926 


0928 
1499 
2069 
2638 
3207 

3775 
4342 
4909 

5474 
6039 

6604 
7167 

773° 
8292 

8853 
9414 
9974 
°533 
1091 
_i649 

2206 
2762 
3318 

3873 
4427 

4980 

5533 
6085 
6636 
7187 


0985 
1556 
2126 
2695 
3264 

3832 

4399 
4965 

553i 
6096 

~666o 
7223 
7786 
8348 
8909 

9470 
°o3o 
0589 

1 147 
1705 

2262 
2818 

3373 
3928 
4482 

5036 
5588 
6140 
6692 

7242 


1042 

1613 

2183 

2752 

33 21 

3888 

4455 
5022 

5587 
6152 


1099 
1670 
2240 
2809 

3377 

3945 
4512 
5078 
5644 
6209 

6773 

733 6 
7898 
8460 
9021 

9582 
°i4i 

0700 
1259 
1816 


1213 

1784 

2354 
2923 

349i 

4°59 
4625 
5192 

5757 
6321 


1271 
1841 
241 1 
2980 
3548 
4115 
4682 
5248 
5813 
6378 
6942 

7505 
8067 
8629 
9190 

9750 

°3°9 

, 0868 

1426 

!9 8 3 
2540 
3096 
3651 
4205 
4759 
5312 

5864 
6416 
6967 
7517 
8067 
8615 
9164 
9711 
0258 

0804 

1349 
1894 

2438 
2981 

3524 
4066 
4607 

5H8 
5688 

6227 
6766 
7304 
7841 
8378 


1328 j 
1898 ! 
2468 
3037 j 
3605 

4172 

4739 
53°5 
5870 

6434 


770 
771 
772 
773 
774 

775 
776 
777 
778 
779 


886491 

7054 
7617 
8179 
8 74I 

9302 

9862 

890421 

0980 
1537 


6547 
7111 

7674 
8236 
8797 

9358 
9918 
0477 
1035 
1593 
2150 
2707 
3262 
3817 
437i 
4925 

5478 
6030 
6581 
7132 

7682 
8231 
8780 
9328 

9875 

0422 
0968 
1513 
2057 
2601 

3*44 

3687 
4229 
4770 

53 10 

5850 
6389 

6927 
7465 
8002 

~«539 

9074 

9610 
0144 
0678 

1211 

1743 
2275 
2806 
3337 
1 


6716 
7280 
7842 
8404 
8965 

9526 
°o86 
0645 
1203 
1760 


6829 
7392 

7955 
8516 
9077 

9638 

°i97 
0756 
1314 
1872 

2429 
2985 

354° 
4094 
4648 

5201 

5754 
6306 

6857 

7407 


6885 

7449 
8011 

8573 
9*34 
9694 

°253 
0812 
1370 
1928 

2484 
3040 

3595 
4150 
4704 

5257 
5809 
6361 
6912 

7462 

8012 
8561 
9109 
9656 

2O3 

0749 
1295 

1840 

2384 

2927 

347° 
4012 

4553 
5094 

5 6 34 
6173 
6712 
7250 

7787 
8324 

8860 
939 6 
993° 
0464 
0998 

153° 
2063 

2594 

3125 

3 6 55 


6998 
7561 
8123 
8685 
9246 

9806 
0365 
0924 
1482 
2039 

2595 
3151 
3706 
4261 
4814 

53 6 7 
5920 
6471 

7022 
7572 


780 
781 
782 
783 
784 

785 
786 
787 
788 
789 


892095 
2651 
3207 
3762 

43 l6 
4870 
5423 
5975 
6526 
7077 

897627 
8176 
8725 

9 2 73 
9821 

900367 
0913 
1458 
2003 
2547 

903090 

3 6 33 
4*74 
4716 

5 2 5 6 
5796 

6 335 
6874 
741 1 
7949 
908485 
9021 

9556 

910091 
0624 

1158 
1690 
2222 
2753 
3284 


2317 
2873 
3429 

3984 
4538 
5091 
5644 
6195 
6747 
7297 


2373 
2929 

3484 
4039 

4593 

5M 6 
5699 
6251 
6802 
_7j52 

7902 
8451 
8999 
9547 
°°94 
0640 
1186 
1731 
2275 
2818 


790 
791 

792 
793 

794 

795 
796 
797 
798 
799 

~800" 
801 
802 
803 
804 

805 

806 
807 
808 
809 


7737 
8286 

8835 
9383 
993° 
0476 
1022 
1567 
2112 
2655 
3199 

374 1 
4283 
4824 
53 6 4 
5904 

6 443 
6981 

7519 
8056 

8592 
9128 
9663 
0197 
0731 
1264 
1797 
2328 
2859 
339° 
2 


7792 
8341 
8890 

9437 
9985 

0531 

1077 
1622 
2166 
2710 

32-53 

3795 
4337 
4878 
5418 

5958 
6497 

7°35 
7573 
8110 

H646 
9181 
9716 

0251 
0784 

1317 
1850 
2381 
2913 
3443 
3 


7847 
8396 

8944 
9492 
0039 

0586 
1131 

1676 
2221 

2764 


7957 
8506 
9054 
9602 
°i 49 . 
0695 
1240 
1785 
2329 
2873 


8122 
8670 
9218 
9766 

0312 

0859 

1404 
1948 
2492 
3036 


33°7 
3849 

439i 
4932 

5472 
6012 
6551 
7089 
7626 
8163 


336i 

39°4 

4445 
4986 

5526 
6066 
6604 

7143 
7680 
8217 


3416 
3958 

4499 
5040 

558o 

6119 

6658 

7196 

7734 
8270 

8807 
9342 

9877 
041 1 
0944 

1477 
2009 
2541 
3072 
3602 

6 


3578 
4120 
4661 
5202 
5742 
6281 
6820 
7358 

7895 
8431 

8967 

95°3 
°°37 
0571 
1 1 04 

1637 
2169 

2700 
3231 
3761 

! 9 


810 
811 
812 
813 
814 

815 
816 
817 
818 

| 819 


8699 

9^35 

9770 
0304 
0838 

1371 
1903 

2435 
2966 
3496 


8753 
9289 
9823 
0358 
0891 

1424 
1956 
2488 
3019 
3549 
5 


8914 
9449 
9984 
o5i8 

1051 

1584 
2116 
2647 

3178 
3708 


P*!- 





4 


7 


8 



31 



N. 


820. 


LOGARITHMS. 




Log. 913. 


N. 
^820' 





1 

786t~ 


2 


3 

3973 


4 


5 


6 

4*3 2 


7 
4184 


8 

4 2 37 


9 


913814 


3920 


4026 


4079 


4290 


821 


4343 


4396 


4449 


4502 


4555 


4608 


4660 


47i3 


4766 


4819 


822 


4872 


4925 


4977 


5030 


5083 


5136 


5189 


5241 


5 2 94 


5347 


823 


5400 


5453 


5505 


5558 


5611 


5664 


5716 


5769 


5822 


5875 


824 


5927 


5980 


6033 


6085 


6138 


6191 


6243 


6296 


6 349 


6401 


825 


6454 


6507 


6559 


6612 


6664 


6717 


6770 


6822 


6875 


6927 


826 


6980 


7033 


7085 


7138 


7190 


7243 


7295 


7348 


7400 


7453 


827 


7506 


7558 


7611 


7663 


7716 


7768 


7820 


7873 


7925 


7978 


828 


8030 


8083 


8i35 


8188 


8240 


8293 


8345 


8397 


8450 


8502 


829 
830 


8555 


8607 
9130 


8659 
9183 


8712 
9235 


8764 


8816 


8869 
939 2 


8921 
9444 


8973 
9496 


9026 
9549 


919078 


9287 


9340 


831 


9601 


9653 


9706 


9758 


9810 


9862 


9914 


9967 


°oi9 


°o7i 


832 


920123 


0176 


0228 


0280 


0332 


0384 


0436 


0489 


0541 


0593 


833 


0645 


0697 


0749 


0801 


0853 


0906 


0958 


IOIO 


1062 


1114 


834 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1530 


1582 


1634 


835 


1686 


1.738 


1790 


1842 


1894 


1946 


1998 


2050 


2102 


2154 


836 


2206 


2258 


2310 


2362 


2414 


2466 


2518 


2570 


2622 


2674 


837 


2725 


2777 


2826 


2881 


2933 


2985 


3°37 


3089 


3140 


3192 


838 


3 2 44 


3296 


3348 


3399 


345i 


35°3 


3555 


3607 


3^58 


3710 


839 


924279 


3814 
433 1 


3865 
4383 


3917 
4434 


3969 


4021 
4538 


4072 
4589 


4* 2 4 
4641 


4176 
4693 


4228 
4744 


840 


4486 


841 


4796 


4848 


4899 


495i 


5003 


5°54 


5106 


5*57 


5209 


5261 


842 


53i 2 


5364 


5415 


5467 


55i8 


557° 


5621 


5 6 73 


5725 


5776 


843 


5828 


5879 


5931 


5982 


6034 


6085 


6137 


6188 


6240 


6291 


844 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 


845 


6857 


6908 


6959 


701 1 


7062 


7114 


7165 


7216 


7268 


7319 


846 


737° 


7422 


7473 


75M 


7576 


7627 


7678 


773° 


7781 


7832 


847 


7883 


7935 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8345 


848 


8396 


8447 


8498 


8549 


8601 


8652 


8703 


8754 


8805 


8857 


849 


8908 


8959 
9470 


9010 


9061 
9572 


9112 


9' 6 3 

9674 


9215 
9725 


9266 

9776 


_9ji7 
9827 


9368 
9879 


850 


929419 


9521 


9623 


851 


993° 


9981 


O032 


°oS3 


°i34 


°i85 


0236 


0287 


0338 


0389 


852 


930440 


0491 


0542 


0592 


0643 


0694 


°745 


0796 


0847 


0898 


853 


0949 


1000 


1051 


1 1 02 


1153 


1204 


1254 


1305 


1356 


1407 


854 


1458 


1509 


1560 


1610 


1661 


1712 


1763 


1814 


1865 


1915 


855 


1966 


2017 


2068 


2118 


2169 


2220 


2271 


2322 


2372 


2423 


856 


2474 


2524 


2575 


2626 


2677 


2727 


2778 


2829 


2879 


2930 


857 


2981 


3031 


3082 


3 J 33 


3183 


3*34 


3285 


3335 


3386 


3437 


858 


3487 


3538 


3589 


3 6 39 


3690 


3740 


379 1 


3841 


3892 


3943 


859 


3993 


4°44 
4549 


4°94 
4599 


4'45 
4650 


4195 


4246 


4296 

4801 


4347 
4852 


4397 
4902 


4448 


860 


934498 


4700 


475i 


4953 


861 


5003 


5°54 


5 io 4 


5154 


5 2 °5 


5*55 


5306 


5356 


5406 


5457 


862 


5507 


5558 


5608 


5658 


5709 


5759 


5809 


5860 


5910 


.5960 


863 


6011 


6061 


6111 


6162 


6212 


6262 


6 3 J 3 


6363 


6413 


6463 


864 


65H 


6564 


6614 


6665 


6715 


6765 


6815 


6865 


6916 


6966 


865 


7016 


7066 


7117 


7167 


7217 


7267 


7317 


7367 


7418 


7468 


866 


7518 


7568 


7618 


7668 


7718 


7769 


7819 


7869 


7919 


7969 


867 


8019 


8069 


8119 


8169 


8219 


8269 


8320 


8370 


8420 


8470 


868 


8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8920 


8970 


869 
~87T 


9020 
939519 


9070 
9569 


9120 
9610 


9*7° 
9669 


9220 


9270 


_932o 
9819 


9369 
9869 


9419 
9918 


9469 


9719 


9769 


9968 


871 


940018 


0068 


0118 


0168 


0218 


0267 


0317 


0367 


0417 


0467 


872 


0516 


0566 


0616 


0666 


0716 


0765 


0815 


0865 


0915 


0964 


873 


1014 


1064 


1114 


. 1163 


1213 


1263 


*3*3 


1362 


1412 


1462 


874 


1511 


1561 


1611 


1660 


1710 


1760 


1809 


1859 


1909 


1958 


875 


2008 


2058 


2107 


2157 


2207 


2256 


2306 


2355 


2405 


2455 


876 


2504 


2554 


2603 


2653 


2702 


2752 


2801 


2851 


2901 


2950 


877 


3000 


3°49 


3099 


3148 


3198 


3 2 47 


3297 


3346 


339 6 


3445 


878 


3495 


3544 


3593 


3643 


3692 


3742 


379i 


3841 


3890 


3939 


879 


3989 


1 


4088 


4H7 
3 


4186 


4 2 3 6 
5 


4285 
6 


_433_5 

7 


4384 
8 


4433 


N. 





2 


4 


9 



32 



N. 


880. 




LOGARITHMS. Log. 


944. 


N. 
880 





1 


2 


3 

4631 


4 


5 


6 


7 


8 


9 


944483 


453* 


4581 


4680 


4729 


4779 


4828 


4877 


4927 


881 


4976 


5° 2 5 


5°74 


5 I2 4 


5*73 


5222 


5272 


53^1 


537o 


54i9 


882 


5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


5862 


5912 


883 


5961 


6010 


6059 


6108 


6157 


6207 


6256 


6 3°5 


6354 


6403 


884 


6452 


6501 


6 55I 


6600 


6649 


6698 


6747 


6796 


6845 


6894 


885 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


733 6 


7385 


886 


7434 


7483 


753 2 


758i 


7630 


7679 


7728 


7777 


7826 


7875 


887 


7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8315 


8364 


888 


8413 


8462 


8511 


8560 


8609 


8657 


8706 


8755 


8804 


8853 


889 


8902 


8951 
9439 


8999 


9048 
9536 


9097 


9146 


9195 
9683 


9244 


9292 
9780 


9341 


890 


949390 


9488 


9585 


9 6 34 


9731 


9829 


891 


9878 


9926 


9975 


O24 


°°73 


°I2I 


O170 


°2I9 


0267 


0316 


892 


95°3 6 5 


0414 


0462 


05 1 1 


0560 


0608 


0657 


0706 


0754 


0803 


893 


0851 


0900 


0949 


0997 


1046 


1095 


"43 


1192 


1240 


1289 


894 


1338 


1386 


H35 


1483 


I53 2 


1580 


1629 


1677 


1726 


1775 


895 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 


i 896 


2308 


2356 


2405 


2 453 


2502 


2 55° 


2599 


2647 


2696 


2744 


j 897 


2792 


2841 


2889 


2938 


2986 


3034 


3 o8 3 


3!3i 


3180 


3228 


1 898 


3276 


33 2 5 


3373 


3421 


347° 


35i8 


3566 


3615 


3663 


37ii 


! 899 


3760 


3808 
4291 


3856 
4339 


3905 
4387 


3953 


4001 


4049 


4098 
4580 


4146 
4628 


4194 

4677 


900 


954243 


4435 


4484 


453 2 


901 


4725 


4773 


4821 


4869 


4918 


4966 


5oi4 


5062 


5110 


5158 


902 


5207 


5255 


53°3 


535i 


5399 


5447 


5495 


5543 


559 2 


5640 


903 


5688 


573 6 


5784 


5832 


5880 


5928 


597 6 


6024 


6072 


6120 


904 


6168 


6216 


6265 


6313 


6361 


6409 


6 457 


6505 


6 553 


6601 


905 


6649 


6697 


6745 


6 793 


6840 


6888 


6936 


6984 


7032 


7080 


906 


7128 


7176 


7224 


7272 


7320 


7368 


7416 


7464 


7512 


7559 


907 


7607 


7655 


77°3 


775 1 


7799 


7847 


7894 


7942 


7990 


8038 


908 


8086 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


8468 


8516 


909 


8564 


8612 
9089 


8659 


8707 
9185 


8755 


8*03 
9280 


8850 
9328 


8898 
9375 


8946 
9423 


8994 


910 


959041 


9137 


9232 


9471 


911 


9518 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 


912 


9995 


O42 


°o9o 


"138 


°i85 


°233 


°28o 


0328 


0376 


°4 2 3 


913 


060471 


o<u8 


0566 
1041 


0613 
1089 


0661 


0709 
1184 


0756 
1231 


0804 
1279 


0851 
1326 


0899 

1374 


914 


0946 


0994 


1136 


915 


1421 


1469 


1516 


1563 


1611 


1658 


1706 


1753 


i8or 


1848 


916 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 


917 


2369 


2417 


2464 


2511 


2 559 


2606 


2653 


2701 


2748 


2795 


918 


2843 


2890 


2937 


2985 


3°3 2 


3°79 


3126 


3 J 74 


3221 


3268 


919 
920 


33* 6 
963788 


3363 

3835 


3410 
3882 


3457 
39 2 9 


35°4 


355 2 


3599 
4071 


3646 
4118 


3 6 93 
4165 


3741 
4212 


3977 


4024 


921 


4260 


43°7 


4354 


4401 


4448 


4495 


4542 


4590 


4 6 37 


4684 


922 


473 1 


4778 


4825 


4872 


4919 


4966 


5013 


5061 


5108 


5155 


923 


5202 


5249 


5296 


5343 


539° 


5437 


5484 


553i 


5.578 


5625 


924 


5672 


5719 


5766 


5813 


5860 


5907 


5954 


6001 


6048 


6095 


925 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6517 


6564 


926 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 


927 


7080 


7127 


7 J 73 


7220 


7267 


73*4 


7361 


7408 


7454 


7501 


928 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 


929 


8016 


8062 


8109 
8576 


8156 
8623 


8203 


8249 


8296 
8763 


8343 
8810 


8390 
8856 


8436 
8903 


930 


968483 


8530 


8670 


8716 


931 


8950 


8996 


9°43 


9090 


9136 


9183 


9229 


9276 


93 2 3 


93 6 9 


932 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 


933 


9882 


9928 


9975 


O2I 


°o68 


ii4 


°i6i 


2O7 


° 2 54 


°3oo 


934 


970347 


°393 


0440 


0486 


°533 


0579 


0626 


0672 


0719 


0765 


935 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 


936 


1276 


1322 


1369 


I415 


1461 


1508 


1554 


1601 


1647 


1693 


937 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


21 10 


2157 


938 


2203 


2249 


2295 


2342 


2388 


2 434 


2481 


2527 


2-573 


2619 


939 


2666 


2712 
1 


2758 


2804 


2851 


2897 
5 


2943 
6 


2989 
7 


3°35 
8 


3082 
9 


N. 





2 


3 


4 



33 



\ = 

N. 


940. 




LOGARITHMS. 




Log. 


973. j 


N. 
940 





1 

3 J 74 


2 


3 

3266 


4 


5 

3359 


6 


7 


8 





973128 


3220 


33 J 3 


3405 


345 1 


3497 


3543 1 


941 


359° 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 


942 


4051 


4097 


4143 


4189 


4 2 35 


4281 


4327 


4374 


4420 


4466 


943 


4512 


455« 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 


944 


4972 


5018 


5064 


5110 


5156 


5202 


5248 


5294 


534° 


5386 


945 


543 2 


5478 


55M 


557o 


5616 


5662 


57°7 


5753 


5799 


5 8 45 


946 


5891 


5937 


59 8 3 


6029 


6075 


6121 


6167 


6212 


6258 


6304 


947 


6350 


6396 


6442 


6488 


6533 


6 579 


6625 


6671 


6717 


6763 


948 


6808 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 


949 


7266 


7312 


735 8 
781s 


74°3 
,7861 


7449 


7495 


754i 
7998 


7586 
8043 


7632 
8089 


7678 
8135 


950 


977724 


7769 


7906 


7952 


951 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8 546 


8591 


952 


8637 


8683 


8728 


8774 


8819 


8865 


8911 


89^6 


9002 


9047 


953 


9°93 


9i3 8 


9184 


9230 


9275 


932i 


9366 


9412 


9457 


9503 


954 


9548 


9594 


9639 


9685 


973° 


9776 


9821 


9867 


9912 


9958 


955 


980003 


0049- 


0094 


0140 


0185 


0231 


0276 


0322 


0367 


0412 


956 


0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 


957 


0912 


0957 


1003 


1048 


1093 


"39 


1 1 84 


1229 


1275 


1320 


958 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 


959 


1819 
982271 


1864 
2316 


!9°9 
2362 


JL954 

2407 


2000 


2045 


2090 
2543 


2135 
2588 


2181 

2633 


2226 
2678 


960 


2452 


2497 


961 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3085 


3130 


962 


3 r 75 


3220 


3265 


3310 


3356 


3401 


344 6 


349i 


353 6 


35 8 i 


963 


3626 


3671 


3716 


3762 


3807 


3 8 52 


3 8 97 


3942 


39 8 7 


4032 


964 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


965 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 


966 


4977 


5022 


5067 


5112 


5157 


5202 


5247 


5292 


5337 


53*2 


967 


5426 


547i 


55i6 


556i 


5606 


5651 


5696 


574i 


57 8 ^ 


5 8 3° 


968 


5875 


5920 


59 6 5 


6010 


.6055 


6100 


6144 


6189 


6234 


6279 


969 
~97ir 


6324 


6369 
6817 


6413 
6861 


6458 
6906 


6503 


6548 


6 593 
7040 


6637 
7° 8 5 


6682 
7130 


6727 


986772 


6951 


6996 


7i75 


971 


7219 


7264 


7309 


7353 


739 8 


7443 


7488 


7532 


7577 


7622 


972 


7666 


7711 


7756 


7800 


7 8 45 


7890 


7934 


7979 


8024 


8068 


973 


8"3 


«M7 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 


974 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 


975 


9005 


9049 


9094 


9138 


9183 


9227 


9272 


9316 


9361 


9405 


976 


9450 


9494 


9539 


95 8 3 


9628 


9672 


9717 


9761 


9806 


9850 


977 


9895 


9939 


9983 


O28 


O072 


°ii7 


0161 


206 


25O 


°294 


978 


990339 


0383 


0428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 


979 
980 


0783 


0827 
1270 


0871 
1315 


0916 
1359 


0960 


1004 


1049 
1492 


1093 
1536 


"37 
1580 


1182 


991226 


1403 


1448 


1625 


981 


1669 


1713 


175 8 


1802 


1846 


1890 


1935 


1979 


2023 


2067 


982 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 


983 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


2951 


984 


2995 


3°39 


3 o8 3 


3127 


3^2 


3216 


3260 


33°4 


334 8 


3392 


985 


343 6 


3480 


35^4 


3568 


3613 


3 6 57 


3701 


3745 


3789 


3 8 33 


986 


3 8 77 


3921 


39 6 5 


4009 


4°53 


4097 


4141 


4185 


4229 


4273 


987 


43*7 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


988 


4757 


4801 


4845 


4889 


4933 


4977 


5021 


5065 


5108 


5 J 52 


989 
990 


5196 
995 6 35 


5 a 4° 

5679 


5284 


532* 
57 6 7 


5372 


541b 
5 8 54 


5460 
5898 


5504 
5942 


5547 
59 8 6 


559i 


5723 


5811 


6030 


991 


6074 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 


6468 


992 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 


993 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 


994 


7386 


7430 


7474 


7517 


7561 


7605 


7648 


7692 


7736 


7779 


995 


7823 


7867 


7910 


7954 


7998 


8041 


8085 


8129 


8172 


8216 


996 


8259 


8 3°3 


8347 


8390 


8434 


8477 


8521 


8^64 


8608 


8652 


997 


8695 


8 739 


8782 


8826 


8869 


8 9*3 


8956 


9000 


9043 


9087 


998 


9131 


9174 


9218 


9261 


93°5 


9348 


9392 


9435 


9479 


9522 


999 


95 6 5 


9609 


9652 


9696 


9739 


97 8 3 


9826 


9 8 7° 
7 


_99^3 
8 


9957 


N. 





1 


2 


3 


4: 


5 


6 


9 



34 



TABLE 



OF 



LOGARITHMIC SINES 



AND 



TANGENTS. 



35 



rr- - "1 

0° LOGARITHMIC 179° 


M. 



Sec. 


Sine. 


Tang. 




60 


M. 
10 


Sec. 


Sine. 


Tang. 




50 I 






7.463725 


7.463727 




10 


5-685575 


5.685575 


50 






10 


70904 


70906 


50 


1 




20 


5.986605 


5 


986605 


40 






20 


77966 


77968 


40 






30 


6.162696 


b 


162696 


30 






30 


84915 


84917 


30 






40 


.287635 




287635 


20 






40 


9 X 754 


7.491756 1 


20 






50 


.384545 




3 8 4545 


10 






50 


7.498487 


7.598490 


JO 




1 




.463726 




463726 




59 


11 




7.505118 


05120 




49 




10 


.530673 




53o673 


50 






10 


1 1 649 


11651 


50 






20 


.588665 




588665 


40 






20 


18083 


18085 


40 






30 


.639817 




639817 


30 






30 


24423 


24426 


30 






40 


•685575 




685575 


20 






40 


30672 


30675 


20 




2 


50 


.726968 




726968 


10 


58 


12 


50 


36832 
42906 


36835 


10 


48 


.764756 




764756 


42909 




10 


.799518 




799518 


50 






10 


48897 


48899 


50 






20 


.831703 




831703 


40 






20 


54806 


54808 


40 






30 


.861666 




861666 


30 






30 


60635 


60638 


30 






40 


.889695 




889695 


20 






40 


66387 


66390 


20 






50 


.916024 




916024 


10 






50 


72065 


72068 


10 




3 




.940847 




940847 




57 


13 




77668 


77671 




47 




10 


.964328 




964329 


50 






10 


83201 


83204 


50 






20 


6.986605 


b 


986605 


40 






20 


88664 


88667 


40 






30 


7.007794 


7 


077794 


30 






30 


94059 


94062 


30 






40 


27998 


27998 


20 






40 


7.599388 


7-599391 


20 




"~ 4 


50 


473°3 
65786 


473°3 


10 


56 


14 


50 


7.604652 


7.604655 


10 


46 


65786 


09853 


09857 




10 


7.083515 


7.083515 


50 






10 


H993 


14996 


50 






20 


7.100548 


7.100548 


40 






20 


20072 


20076 


40 






30 


16938 


16939 


30 






30 


25093 


25097 


30 






40 


3 2 733 


3 2 733 


20 






40 


3C056 


30060 


20 






50 


47973 


47973 


10 






50 


34963 


34968 


10 




5 




1^696 


62696 




55 


13 




3 c?:6 


^C.?20 




45 




10 


76936 


76937 


50 






10 


44615 


44 6j 9 


50 






20 


7.190725 


7.190725 


40 






20 


49361 


49366 


40 






30 


7.204089 


7.204089 


30 






30 


54056 


54061 


30 






40 


17054 


17054 


20 






40 


58701 


58706 


20 




6 


50 


29643 


29643 

41878 


10 


54 


16 


50 


63297 
67845 


633OI 
67849 


10 


44 


41877 




10 


53776 


53777 


50 






10 


7 2 345 


72350 


50 






20 


65358 


6 5359 


40 






20 


76799 


76804 


40 






30 


76639 


76640 


30 






30 


81208 


81213 


30 






40 


87635 


87635 


20 




■ 


40 


8 5573 


85578 


20 






50 


7.298358 


7.298359 


10 






60 


89894 


899OO 


10 




7 




7.308824 


7.308825 




53 


17 




94173 


94*79 




43 




10 


19043 


19044 


50 






10 


7.698410 


7.698416 


50 






20 


29027 


29028 


40 






20 


7.702606 


7.702612 


40 






30 


38787 


38788 


30 






30 


06762 


06768 


30 






40 


■ 4 8 33 2 


48333 


20 






40 


10879 


10885 


20 




8 


50 


57672 


57673 
66817 


10 


52 


18 


50 


H957 
18997 


14962 
19003 


10 


42 


66816 




10 


75770 


75772 


50 






10 


22999 


23005 


50 






20 


84544 


84546 


40 






20 


26966 


26972 


40 






30 


7-393 x 45 


7.393146 


30 






30 


30896 


30902 


30 






40 


7.401578 


7.401579 


20 






40 


34791 


34797 


20 






50 


09850 


09852 


10 






50 


38651 


38658 


10 




9 




17968 


17970 




51 


19 




4H77 


42484 




41 




10 


25937 


2 5939 


50 






10 


46270 


46277 


50 






20 


33762 


33764 


40 






20 


50031 


5oo37 


40 






30 


41449 


41451 


30 






30 


53758 


53765 


30 






40 


49002 


49004 


20 






40 


57454 


57462 


20 






50 


56426 


56428 


10 






50 


61119 


61127 


10 




10 




7.463725 
Cosine. 


7.463727 


Sec. 


50 

M. 


20 




7-764754 
Cosine. 


7.764761 
Cotang. 


Sec. 


40 

M. 


Cotang. 


90° 


89° 



36 



0° 


SX3HSS 


A2JTO 


TAKTCKE2MTS 


■ 


; 

179° 


I M. 
20 


Sec. 


Sine. 


Tang. 




40 


M. 
~30 


Sec. 


Sine. 


Tang. 




i 
30 


7.764754 


7.764761 




7.940842 


7.940858 




10 


68358 


68365 


50 






10 


43248 


43 26 5 


50 






20 


71932 


71940 


40 






20 


45641 


45657 


40 






30 


75477 


754 8 5 


30 






30 


48020 


48037 


30 






40 


78994 


79002 


20 






40 


50387 


50404 


20 






50 


82482 


82490 


10 






50 


52741 


5 2 758 


10 




21 




85943 


85951 




39 


31 




55082 


55100 




^9 




10 


89376 


89384 


50 






10 


574i° 


57428 


50 






20 


92782 


92790 


40 






20 


59727 


59745 


40 






30 


96162 


96170 


30 






30 


62031 


62049 


30 






40 


7-7995*5 


7.799524 


20 






40 


64322 


64341 


20 




22 


50 


7.802843 


7.802852 
06155 


10 


38 


32 


50 


66602 


66621 


10 


28 




06146 


68870 


68889 




10 


09423 


09432 


50 






10 


71126 


71145 


50 






20 


12677 


12686 


40 






20 


73370 


73389 


40 






30 


i59°5 


I59I5 


30 






30 


75603 


75622 


30 






40 


19111 


19120 


20 






40 


77824 


77844 


20 






50 


22292 


22302 


10 






50 


80034 


80054 


10 




23 




25451 


25460 




37 


33 




82233 


82253 




27 




10 


28586 


28596 


50 






10 


84421 


84441 


50 






20 


31700 


31710 


40 






20 


86598 


86618 


40 






30 


34791 


34801 


30 






30 


88764 


88785 


30 






40 


37860 


37870 


20 






40 


90919 


90940 


20 




24 


50 


40907 


40918 


10 


36 


34 


50 


93064 


93085 


10 


26 


43934 


43944 


95198 


95219 




10 


46939 


46950 


50 






10 


97322 


97343 


50 






20 


49924 


49935 


40 






20 


7-999435 


7.999456 


40 




1 


30 


52888 


52900 


30 






30 


8.001538 


8.001560 


30 






40 


55833 


55844 


20 






40 


03631 


03653 


20 






50 


5 8 757 


58769 


10 






50 


05714 


05736 


10 




25 




61662 


61674 




35 


35 




07787 


07809 




25 




10 


64548 


64560 


50 






10 


09850 


09872 


50 






20 


67414 


67426 


40 






20 


11903 


11926 


40 






30 


70262 


70274 


30 






30 


13947 


13970 


30 






40 


73092 


73104 


20 






40 


15981 


16004 


20 




26 


50 


75902 


75915 


10 


34 


36 


50 


18005 


18029 


10 


24 


78695 


78708 


20021 


20044 




10 


81470 


81483 


50 






10 


22027 


22051 


50 






20 


84228 


84240 


40 






20 


24023 


24047 


40 






30 


86968 


86981 


30 






30 


26011 


26035 


30 






40 


89690 


89704 


20 






40 


27989 


28014 


20 






50 


92396 


92410 


10 






50 


29959 


29984 


10 




27 




95085 


95099 




33 


37 




3^9 


3*945 




23 




10 


7.897758 


7.897771 


50 






10 


33 8 7i 


33897 


50 






20 


7.900414 


7.900428 


40 






20 


35814 


35840 


40 






30 


°3°54 


03068 


30 






30 


37749 


37775 


30 






40 


05678 


05692 


20 






40 


39675 


39701 


20 




28 


50 


08287 


08301 


10 


32 


38 


50 


41592 


41618 


10 


22 


10879 


10894 


43501 


43527 




10 


13457 


13471 


50 






10 


45401 


45428 


50 






20 


16019 


16034 


40 






20 


47294 


473 21 


40 






30 


18566 


18581 


30 






30 


49178 


49205 


30 






40 


21098 


21113 


20 






40 


51054 


51081 


20 






50 


23616 


23631 


10 






50 


52922 


5 2 949 


10 




29 




26119 


26134 




31 


39 




54781 


54809 




21 




10 


28608 


28623 


50 






10 


56633 


56661 


50 






20 


31082 


31098 


40 






20 


58477 


58506 


40 






30 


33543 


33559 


30 






30 


60314 


60342 


30 






40 


359 8 9 


36006 


20 






40 


62142 


62171 


20 






50 


38422 


3 8 439 


10 






50 


63963 


63992 


10 




30 




7.940842 


7.940858 


Sec. 


30 

M. 


40 




8.065776 


8.065806 


Sec. 


20 

M. 


Cosine. 


Cotang. 


Cosine. 


Cotang. 


90° 


















89° 



24 



37 





0° 




LOGARITHIVIIC 


179° 




M. 
40 


Sec. 


Sine. 
8.065776 


Tang. 




20 


M. 
50 


Sec. 


Sine. 


Tang. 




10 




8.065806 


8.162681 


8.162727 






10 


67582 


67612 


50 






10 


64126 


64172 


50 








20 


69380 


69410 


40 






20 


65566 


65613 


40 








30 


71171 


71201 


30 






30 


67002 


67049 


30 








40 


7 2 955 


72985 


20 






40 


68433 


68480 


20 








50 


7473 1 


74761 


10 






50 


69859 


69906 


10 






41 




76500 


76531 




19 


51 




71280 


71328 




9 






10 


78261 


78293 


50 






10 


72697 


72745 


50 








20 


80016 


80047 


40 






20 


74109 


74158 


40 








30 


81764 


81795 


30 






30 


75517 


75566 


30 








40 


83504 


83536 


20 






40 


76920 


76969 


20 






42 


50 


85238 


85270 


10 


18 


52 


50 


78319 


78368 


10 


8 




86965 


86997 


79713 


79763 






10 


88684 


88717 


50 






10 


81102 


81152 


50 








20 


90398 


90430 


40 






20 


82488 


82538 


40 








30 


92104 


92137 


30 






30 


83868 


83919 


30 








40 


93804 


93 8 37 


20 






40 


85245 


85296 


20 








50 


95497 


• 9553° 


10 






50 


86617 


86668 


10 






43 




97183 


97217 




17 


53 




87985 


88036 




7 






10 


8.098863 


8.098897 


50 






10 


89348 


89400 


50 








20 


8.100537 


8.100571 


40 






20 


90707 


90760 


40 








30 


02204 


02239 


30 






30 


92062 


92115 


30 








40 


03864 


03899 


20 






40 


93413 


93466 


20 






44 


50 


05519 


°5554 


10 


16 


54 


50 


94760 
96102 


94813 


10 






07167 


07202 


96156 




6 






10 


08809 


08845 


50 






10 


97440 


97494 


50 








20 


10444 


1 048 1 


40 






20 


8.198774 


8.198829 


40 








30 


12074 


12110 


30 






30 


8.200104 


8.200159 


30 








40 


13697 


13734 


20 






40 


01430 


01485 


20 








50 


153*5 


15352 


10 






50 


02752 


02808 


10 






45 




16926 


16963 




15 


55 




04070 


04126 




5 






10 


18532 


18569 


50 






10 


05384 


05440 


50 








20 


20131 


20169 


40 






20 


06694 


06750 


40 








30 


21725 


21763 


30 






30 


08000 


08057 


30 








40 


23313 


23351 


20 






40 


09302 


09359 


20 






46 


50 


24895 


24933 


10 


14 


56 


50 


10601 


10658 
11953 


10 


4 




8.126471 


8.126510 


11895 






10 


28042 


28081 


50 






10 


I3I85 


13243 


50 








20 


29606 


29646 


40 






20 


14472 


14530 


40 








30 


31166 


31206 


30 






30 


15755 


15S14 


30 








40 


32720 


32760 


20 






40 


17034 


17093 


20 








50 


34268 


34308 


10 






50 


18309 


18369 


10 






47 




35810 


35851 




13 


57 




8.219581 


8.219641 




3 






10 


37348 


37389 


50 






10 


20849 


20909 


50 








20 


38879 


38921 


40 






20 


22113 


22174 


40 








30 


40406 


40447 


30 






30 


23374 


23434 


30 








40 


41927 


41969 


20 






40 


24631 


24692 


20 






48 


50 


43443 
44953 


43485 
44996 


10 


12 


58 


50 


25884 


25945 
27195 


10 


2 




27133 






10 


46458 


46501 


50 






10 


28380 


28442 


50 








20 


47959 


48001 


40 






20 


29622 


29685 


40 








30 


49453 


49497 


30 






30 


30861 


30924 


30 








40 


5°943 


50987 


• 20 






40 


32096 


T2l6o 


20 








50 


52428 


52472 


10 






50 


33328 


33392 


10 






49 




539°7 


53952 




11 


59 




34557 


34621 




1 






10 


553 82 


55426 


50 






10 


35782 


35846 


50 








20 


56852 


56896 


40 






20 


37003 


37068 


40 








30 


58316 


58361 


30 






30 


38221 


38286 


30 








40 


59776 


59821 


20 






40 


39436 


39501 


20 


■ 






50 


61231 


61276 


10 






50 


40647 


40713 


10 






50 


i 


8.162681 


8.162727 


Sec. 


10 

M. 


60 




8.241855 


8.241921 
Cotang. 


Sec. 


O 

M. 




Cosine. 


Cotang. 


Cosine. 




90° 










89° 



38 













1: 


1° 


SINES AND 


TANGENTS. 


178° 


M. 



Sec. 


Sine. 


Tang. 
8.241921 




60 


M. 
10 


Sec. 


Sine. 


Tang. 




50 


8.241855 


8.308794 


8.308884 




10 


3060 


3126 


50 






10 


8.309827 


8.309917 


50 






20 


4261 


4328 


40 






20 


8.310857 


8.310948 


40 






30 


5459 


5526 


30 






30 


1885 


1976 


30 






40 


6654 


6721 


20 






40 


2910 


3002 


20 




50 


7845 


7913 


10 






50 


3933 


4025 


10 


1 


1 




8.249033 


8.249101 




59 


11 




4954 


5046 




49 




10 


8.250218 


8.250287 


50 






10 


5972 


6065 


50 






20 


1400 


1469 


40 






20 


6987 


7081 


40 






30 


2578 


2648 


30 






30 


8001 


8095 


30 






40 


3753 


3823 


20 






40 


8.319012 


8.319106 


20 




2 


50 


4925 


4996 


10 


58 


12 


50 


8.320021 


8.320115 
1122 


10 


48 


6094 


6165 


1027 




10 


7260 


733 1 


50 






10 


2031 


2127 


50 






20 


8423 


8494 


40 






20 


3°33 


3129 


40 






30 


8.259582 


8.259654 


30 






30 


4032 


4128 


30 






40 


8.260739 


8.260811 


20 






40 


5029 


5126 


20 






50 


1892 


1965 


10 






50 


6024 


6l2I 


10 




3 




3042 


3 JI 5 




57 


13 




7016 


7114 




47 




10 


4190 


4263 


50 






10 


8007 


8105 


50 






20 


5334 


5408 


40 






20 


8995 


8.329093 


40 






30 


6475 


6549 


30 






30 


8.329980 


8.330080 


30 






40 


7613 


7688 


20 






40 


8.330964 


I064 


20 




4 


50 


8749 


8824 


10 


56 


14 


50 


1945 
2924 


2045 

3°25 


10 


46 


8.269881 


8.269956 




10 


8.271010 


8.271086 


60 






10 


3901 


4002 


50 






20 


2137 


2213 


40 






20 


4876 


4977 


40 






30 


3260 


3337 


30 






30 


5848 


595° 


30 






40 


438i 


4458 


20 






40 


6819 


6921 


20 






50 


5499 


5576 


10 






50 


7787 


7890 


10 




5 




6614 


6691 




55 


15 




8753 


8856 




45 




10 


7726 


7804 


50 






10 


8.339717 


8.339821 


50 






20 


8835 


8.278913 


40 






20 


8.340679 


8.340783 


40 






30 


8.279941 


8.280020 


30 






30 


1638 


1743 


30 






40 


8.281045 


1 1 24 


20 






40 


2596 


2701 


20 




6 


50 


2145 


2225 


10 


54 


16 


50 


355i 


3657 
4610 


10 


44 




3*43 


33*3 


4504 




10 


4339 


4419 


60 






10 


5456 


5562 


50 






20 


543i 


55 1 * 


40 






20 


6405 


6512 


40 






30 


6521 


6602 


30 






30 


7352 


7459 


30 






40 


7608 


7689 


20 






40 


8297 


8405 


20 






50 


8692 


8774 


10 






50 


8.349240 


8.349348 


10 




7 




8.289773 


8.289856 




53 


17 




8.350181 


8.350289 




43 




10 


8.290852 


8.290935 


60 






10 


1119 


1229 


50 






20 


1928 


2012 


40 






20 


2056 


2166 


40 






30 


3002 


3086 


30 






30 


2991 


3101 


30 






40 


4°73 


4i57 


20 






40 


39M 


4°35 


20 




8 


50 


5141 


5226 


10 


52 


18 


50 


4855 


4966 
5895 


10 


42 


6207 


6292 


5783 




10 


7270 


7355 


50 






10 


6710 


6823 


50 






20 


8330 


8416 


40 






20 


7635 


7748 


40 






30 


8.299388 


8.299474 


30 






30 


8558 


8671 


30 






40 


8.300443 


8.300530 


20 






40 


8.359479 


8-359593 


20 






50 


1496 


1583 


10 






50 


8.360398 


8.360512 


10 




9 




2546 


2633 




51 


19 




1315 


1430 




41 




10 


3594 


3682 


50 






10 


2230 


2345 


50 






20 


4639 


4727 


40 






20 


3H3 


3259 


40 






30 


5681 


5770 


30 






30 


4054 


4171 


30 






40 


6721 


6811 


20 






40 


4964 


5080 


20 






60 


7759 


7849 


10 






50 


5871 


5988 


10 




10 




8.308794 


8.308884 


Sec. 


50 

M. 1 


20 




8.366777 


8.366894 
Cotang. 


Sec. 


40 

M. 


Cosine. 


Cotang. 


Cosine. 


91° 










88° 



39 



I ■ 

1° 




LOGARITHZVZIC 


178° 


1 M. 
~20 


Sec. 


Sine. 


Tang. 




40 


M. 
30 


Sec. 


Sine. 


Tang. 




30 


8.366777 


8.366894 


8.417919 


8.418068 




10 


7681 


7799 


50 






10 


8722 


8872 


50 






20 


8582 


8701 


40 






20 


8.419524 


8.419674 


40 






30 


8.369482 


8.369601 


30 






30 


8.420324 


8.420475 


30 






40 


8.370380 


8.370500 


20 






40 


1123 


1274 


20 






50 


1277 


1397 


10 






50 


1921 


2072 


10 




21 




2171 


2291 




39 


31 




2717 


2869 




29 




10 


3063 


3184 


50 






10 


35" 


3664 


50 






20 


3954 


4076 


40 






20 


4304 


4458 


40 






30 


4 8 43 


4965 


30 






30 


5096 


5250 


30 






40 


573° 


5853 


20 






40 


5886 


6040 


20 




22 


50 


6615 
7499 


6738 
7622 


10 


38 


32 


50 


6675 


6830 
7618 


10 


28 


7462 




10 


8380 


8504 


50 






10 


8248 


8404 


50 






20 


8.379260 


8-379385 


40 






20 


9032 


9189 


40 






30 


8.380138 


8.380263 


30 






30 


8.429815 


8.429973 


30 






40 


1015 


1 140 


20 






40 


8.430597 


8-43°755 


20 






50 


1889 


2015 


10 






50 


1377 


I53 6 


10 




23 




2762 


2889 




37 


33 




2156 


2315 




27 




10 


3 6 33 


3760 


50 






10 


2933 


3°93 


50 






20 


4502 


4630 


40 






20 


3709 


3870 


40 






30 


537o 


5498 


30 






30 


4484 


4645 


30 






40 


6236 


6364 


20 






40 


5 2 57 


5419 


20 




24 


50 


7100 
7962 


7229 


10 


36 


34 


50 


6029 
6800 


6191 


10 




8092 


6962 




26 




10 


8823 


8953 


50 






10 


7569 


773 2 


50 






20 


8.389682 


8.389812 


40 






20 


8337 


8500 


40 






30 


8-39°539 


8.390670 


30 






30 


9103 


8.439267 


30 






40 


1395 


1526 


20 






40 


8.439868 


8.440033 


20 






50 


2249 


2381 


10 






50 


8.440632 


0797 


10 




25 




3101 


3234 




35 


35 




1394 


1560 




25 




10 


395i 


4085 


50 






10 


2155 


2322 


50 






20 


4800 


4934 


40 






20 


2915 


3082 


40 






30 


5 6 47 


5782 


30 






30 


3 6 74 


3841 


30 






40 


6 493 


6628 


20 






40 


4431 


4599 


20 




26 


50 


7337 
8179 


7472 


10 


34 


36 


50 


5186 


5355 


10 


24 


8315 


594i 


6110 




10 


9020 


9156 


50 






10 


6694 


6864 


50 






20 


8 -399 8 59 


8.399996 


40 






20 


7446 


7616 


40 






30 


8.400696 


8.400834 


30 






30 


8196 


8367 


30 






40 


I53 2 


1670 


20 






40 


8946 


9117 


20 






50 


2366 


2505 


10 






50 


8.449694 


8.449866 


10 




27 




3199 


3338 




33 


37 




8.450440 


8.450613 




23 




10 


4030 


4170 


50 






10 


1186 


1359 


50 






20 


4859 


5000 


40 






20 


1930 


2104 


40 






30 


5687 


5828 


30 






30 


2672 


2847 


30 






40 


6513 


6655 


20 






40 


3414 


3589 


20 




28 


50 


733 8 
8161 


7480 
8304 


10 


32 


38 


50 


4154 


433° 


10 


22 


4893 


5070 




10 


8983 


9126 


50 






10 


5 6 3! 


5808 


50 




■ 


20 


8.409803 


8.409946 


40 






20 


6368 


6545 


40 






30 


8.410621 


8.410765 


30 






30 


7103 


7281 


30 






40 


1438 


I58-3 


20 






40 


7837 


8016 


20 






50 


2254 


2399 


10 






50 


8570 


8749 


10 




29 




3068 


3213 




31 


39 




8.459301 


8.459481 




21 




10 


3880 


4026 


50 






10 


8.460032 


8.460212 


50 






20 


4691 


4837 


40 






20 


0761 


0942 


40 






30 


55°° 


5 6 47 


30 






30 


1489 


1670 


30 






40 


6308 


6456 


20 






40 


2215 


2398 


20 






50 


7114 


7262 


10 






50 


2941 


3124 


10 




30 




8.417919 


8.418068 


Sec. 


30 

M. 


40 




8.463665 


8.463849 


Sec. 


20 

M. 




Cosine. 


Cotang. 


Cosine. 


Cotang. 


| 91° 










88° 



40 



1° 


SINES AND 


TANGENTS 


178° 


M. 
40 


Sec. 


Sine. 


Tang. 




20 


M. 


Sec. 


Sine. 


Tang. 




To 


8.463665 


8.463849 


50 




8.505045 


8.505267 




10 


4388 


4572 


50 






10 


5702 


592.5 


50 






20 


5110 


52-95 


40 






20 


6358 


6582 


40 






30 


5830 


6016 


30 






30 


7014 


7238 


30 






40 


655 


6736 


20 






40 


7668 


7893 


20 






50 


7268 


7455 


10 






50 


8321 


8547 


10 




41 




7985 


8172 




19 


51 




8974 


9200 




9 




10 


8701 


8889 


50 






10 


8.509625 


8.509852 


50 






20 


8.469416 


8.469604 


40 






20 


8.510275 


8.510503 


40 






30 


8.470129 


8.470318 


30 






30 


0925 


II53 


30 






40 


0841 


1031 


20 






40 


1573 


1802 


20 




42 


50 


1553 


1743 


10 


18 


52 


50 


2221 


2451 

3098 


10 


"s~ 


2263 


2454 


2867 




10 


2971 


3 l6 3 


50 






10 


3513 


3744 


50 






20 


3679 


3871 


40 






20 


4157 


43 8 9 


40 






30 


4386 


4579 


30 






30 


480I 


5°34 


30 






40 


5 9! 


5285 


20 






40 


5444 


5677 


20 






50 


5795 


5990 


10 






50 


6086 


6319 


10 




43 




6498 


6693 




17 


53 




6726 


6961 




7 




10 


7200 


7396 


50 






10 


7366 


7602 


50 






20 


7901 


8097 


40 






20 


8005 


S241 


40 






30 


8601 


8798 


30 






30 


8643 


8880 


30 






40 


9299 


8.479497 


20 






40 


9280 


8.519517 


20 




44 


50 


8.479997 


8.480195 


10 


16 


54 


50 


8.519916 
8.520551 


8.520154 
0790 


10 


~6~ 


8.480693 


0892 




10 


1388 


1588 


50 






10 


1186 


1425 


50 






20 


2082 


2283 


40 






20 


1819 


2059 


40 






30 


2775 


2976 


30 






30 


2451 


2692 


30 






40 


34 6 7 


3669 


20 






40 


3 o8 3 


33 2 4 


20 






50 


4158 


4360 


10 






50 


3713 


3956 


10 




45 




4848 


5°5° 




15 


55 




4343 


4586 




5 




JLO 


553 6 


574° 


50 






10 


4972 


5 2 i5 


50 






20 


6224 


6428 


40 






20 


5599 


5844 


40 






30 


6910 


7115 


30 






30 


6226 


6472 


30 






40 


7596 


7801 


20 






40 


6852 


7098 


20 




46 


50 


8280 


8486 


10 


14 


56 


50 


7477 
8102 


7724 


10 


4 


8963 


9170 


8349 




10 


8.489645 


8.489852 


50 






10 


8725 


8973 


50 






20 


8.490326 


8.490534 


40 






20 


9347 


8.529596 


40 






30 


1006 


1215 


30 






30 


8.529969 


8.530218 


30 






40 


1685 


1894 


20 






40 


8.530589 


0840 


20 






50 


2363 


2573 


10 






50 


1209 


1460 


10 




47 




3040 


3250 




13 


57 




1828 


2080 




3 




10 


37i5 


39 2 7 


50 






10 


2446 


2698 


50 






20 


439° 


4602 


40 






20 


3 o6 3 


3316 


40 






30 


5064 


5276 


30 






30 


3679 


3933 


30 






40 


573 6 


5949 


20 






40 


4295 


4549 


20 




48 


50 


6408 
7078 


6622 


10 


12 


58 


50 


4909 
5523 


5164 
5779 


10 


2 


7293 




10 


7748 


79 6 3 


50 






10 


6136 


6392 


50 






20 


8416 


8632 


40 






20 


6747 


7005 


40 






30 


9084 


9300 


30 






30 


7358 


7616 


30 






40 


8.499750 


8.499967 


20 






40 


7969 


8227 


20 






50 


8.500415 


8.500633 


10 






50 


8578 


8837 


10 




49 




1080 


1298 




11 


59 




9186 


8-539447 




1 




10 


1743 


1962 


50 






10 


8-539794 


8.540055 


50 






20 


2405 


2625 


40 






20 


8.540401 


0662 


40 






30 


3067 


3287 


30 






30 


1007 


1269 


30 






40 


3727 


3948 


20 






40 


1612 


1875 


20 






50 


4386 


4608 


10 






50 


2216 


2480 


10 




50 




8.505045 


8.505267 


Sec. 


10 

M. 


60 


— 


8.542819 
Cosine. 


8.543084 
Cotang. 


Sec. 


O 

M. 






Cosine. 


Cotang. 


91° 








88° 



41 



0° 




LOGARITHMIC 


1 


179° 




M. 

~~6 


Sine. 


Diff. 1" 


Cosine. 


Diff. 1" 


Tan-. 


Diff. 1" 


Cotang. 


1 

1 

60 




Inf. neg. 1 




10.000000 


.00 


Inf. neg. 




Infinite. 




1 


6.463726 


5017.17 







6.463726 


5017.17 


13.536274 


59 




2 


764756 


2934.85 


O 




764756 


2934.85 


235244 


58 




'6 


6.940847 


2082.31 







6.940847 


2082.31 


i3- 59 J 53 


57 




4 


7.065786 


1615.17 







7.065786 


1615.17 


12.934214 


56 




5 


162696 


1319.68 





.00 


162696 


1319.69 


837304 


55 




6 


241877 


III5.78 


9.999999 


.01 


241878 


1115.78 


758122 


54 




V 


308824 


966.53 


99 




308825 


966.54 


691175 


53 




8 


366816 


852.54 


99 




366817 


852.54 


633183 


52 




y 


417968 


762.62 


99 




417970 


762.63 


582030 


51 




10 


463725 


689.88 


98 

9.999998 




463727 


689.88 


536273 


50 
49 




7.505118 


629.81 


7.505120 


629.81 


12.494880 




12 


542906 


579-3 6 


97 




542909 


579-37 


457091 


48 




13 


577668 


53 6 -4* 


97 




577672 


536.42 


422328 


47 




14 


609853 


499.38 


9 6 




609857 


499-39 


390143 


46 




15 


639816 


467.14 


96 




639820 


467.15 


360180 


45 




16 


667845 


438.81 


95 




667849 


438.82 


332151 


44 




17 


694173 


4I3-7 2 


95 




694179 


4*3-73 


305821 


43 




18 


718997 


39 I -35 


94 




719003 


391.36 


280997 


42 




19 


742477 


37!- 2 7 


93 




742484 


371.28 


257516 


41 




20 
21 


764754 


353- x 5 


93 




764761 
7.785951 


353.16 


235239 


40 
39 




7-785943 


336.72 


9.999992 


33 6 -73 


12.214049 




22 


806146 


3 2I -75 


9 1 




806155 


321.76 


193845 


38 




23 


825451 


308.05 


90 


.01 


825460 


308.07 


174540 


37 




24 


843934 


295.47 


89 


.02 


843944 


295.49 


156056 


36 




2o 


861662 


283.88 


88 




861674 


283.90 


138326 


35 




26 


878695 


273.17 


88 




878708 


273.18 


121292 


34 




27 


895085 


263.23 


87 




895099 


263.25 


104901 


33 




28 


910879 


253-99 


86 




910894 


254.01 


089106 


32 




29 


, 926119 


245.38 


85 




926134 


245.40 


073866 


31 




30 
31 


940842 


237-33 
229.80 


83 
9.999982 




940858 


237-35 
229.82 


059142 


30 
29 




7.955082 




7.955100 


12.044900 




32 


968870 


222.73 


81 




968889 


222.75 


031111 


28 




33 


982233 


216.08 


80 




982253 


216.10 


017747 


27 




34 


7.995198 


209.81 


79 




7.995219 


209.83 


12.004781 


26 




35 


8.007787 


203.90 


77 




8.007809 


203.92 


11.992191 


25 




36 


020021 


198.31 


76 




020045 


I98-33 


979955 


24 




37 


031919 


193.02 


75 




031945 


193.05 


968055 


23 




38 


043501 


188.01 


73 




043527 


188.03 


956473 


22 




39 


054781 


183.25 


72 




054809 


183.27 


945191 


21 




40 
41 


065776 
8.076500 


178.72 


7i 




065806 


178.75 


934194 


20 
19 




174.41 


9.999969 




8.076531 


174.44 


11.923469 




42 


086965 


170.31 


68 




086997 


170.34 


913003 


18 




43 


097183 


166.39 


66 


.02 


097217 


166.42 


902783 


17 




44 


107167 


162.65 


64 


.03 


107202 


162.68 


892798 


16 




45 


116926 


159.08 


63 




1 16963 


159.11 


883037 


15 




46 


126471 


155.66 


61 




126510 


155.68 


873490 


14 




47 


135810 


152.38 


59 




135851 


152.41 


864149 


13 




48 


144953 


149.24 


58 




144996 


149.27 


855004 


12 




49 


153907 


146.22 


56 




153952 


146.25 


846048 


11 




50 
51 


162681 


H3-33 
140.54 


54 
9.999952 




162727 


H3-3 6 

140.57 


837273 
11.828672 


10 
9 




8.171280 


8.171328 




52 


179713 


137.86 


5° 




179763 


137.90 


820237 


8 




53 


187985 


135.29 


48 




188036 


I35-32 


811964 


7 




54 


196102 


132.80 


46 




196156 


132.84 


803844 


6 




55 


204070 


130.41 


44 


.03 


204126 


130.44 


795874 


5 




56 


211895 


128.10 


42 


.04 


211953 


128.14 


788047 


4 




57 


219581 


125.87 


40 




219641 


125.91 


780359 


3 




58 


227134 


123.72 


38 




227195 


123.76 


772805 


2 




59 


234557 


121.64 


36 


.04 


234621 


121.68 


765379 


1 




60 


8.241855 




9-999934 

Sine. 




8.241921 




11.758079 



M. 




Cosine. 


Diff. 1" 


Diff.1" 


Cotang. 


Diff. 1" 


Tang. 




-90° 








89° 





42 



! r 


> 


SINES AND TANGENTS. 


178° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 
.04 


Tang. 


Diff. 1" 


Cotang. 


60 


8.241855 


119.63 


9-999934 


8.241921 


II9.67 


II.758079 


1 


249033 


117.68 


93 2 




249102 


117.72 


750898 


59 


2 


256094 


115.80 


929 




256165 


115.84 


743835 


58 


8 


263042 


113.98 


927 




263115 


114.02 


736885 


57 


4 


269881 


112. 21 


925 




269956 


112,25 


730044 


56 


5 


276614 


HO.50 


922 




276691 


IIO.54 


7233 9 


55 


6 


283243 


I08.83 


920 




283323 


108.87 


716677 


54 


V 


289773 


IO7.22 


917 




289856 


107.26 


710144 


53 


8 


296207 


IO5.65 


9*5 




296292 


105.70 


703708 


52 


9 


302546 


IO4.I3 


912 




302634 


1 04. 1 8 


697366 


51 


10 
11 


308794 
8.314954 


I02.66 
IOI.22 


910 




308884 


102.70 


691116 


50 
49 


9.999907 


8.315046 


101.26 


11.684954 


12 


321027 


99.82 


905 




321122 


99.87 


678878 


48 


13 


327016 


98.47 


902 


.04 


327114 


98.51 


672886 


47 


14 


332924 


97.I4 


899 


.05 


333° 2 5 


97.19 


* 666975 


46 


16 


33 8 753 


95 .86 


897 




338856 


95.90 


661144 


45 


16 


3445°4 


94.60 


894 




344610 


94.65 


655390 


44 


IV 


350180 


93-3 8 


891 




350289 


93-43 


64971 1 


43 


18 


355783 


92.19 


888 




355895 


92.24 


644105 


42 


iy 


361315 


91.03 


885 




361430 


91.08 


638570 


41 


20 
21 


366777 
8.372171 


89.90 


882 




366895 


89.95 
88.85 


, 633105 


40 
39 


88.80 


9.999879 


8.372292 


II.627708 


22 


377499 


87.72 


876 




377622 


87.77 


622378 


38 


23 


382762 


86.67 


873 




382889 


86.72 


617111 


37 


24 


387962 


85.64 


870 




388092 


85.70 


611908 


36 


25 


393101 


84.64 


867 




393 2 34 


84.69 


606766 


35 


26 


398179 


83.66 


864 




398315 


83-71 


601685 


34 


27 


403199 


82.71 


861 




403338 


82.76 


596662 


33 


28 


408 1 61 


81.77 


858 




408304 


81.82 


591696 


32 


29 


413068 


80.86 


854 


.05 


41321^ 


80.91 


586787 


31 


30 
31 


417919 
8.422717 


79.96 
79.09 


851 


.06 


418068 


80.02 


581932 
11.577131 


30 
29 


9.999848 




8.422869 


79.14 


32 


427462 


78.23 


844 




427618 


78.29 


572382 


28 


33 


432156 


77.40 


841 




43 2 3!5 


77-45 


567685 


27 


34 


436800 


76.57 


838 




436962 


76.63 


563038 


26 


35 


441394 


75-77 


834 




441560 


75-83 


558440 


25 


36 


445941 


74-99 


831 




446 1 1 


75-°5 


553890 


24 


3/ 


450440 


74.22 


827 




450613 


74.28 


549387 


23 


38 


454 8 93 


73-4 6 


823 




455070 


73-52 


54493° 


22 


39 


4593oi 


72.73 


820 




459481 


72.79 


540519 


21 


40 
41 


463665 
8.467985 


72.00 


816 
9.999813 




463849 
8.468173 


72.06 


536151 


20 
19 


71.29 




71-35 


II. 531827 


42 


472263 


70.60 


809 




472454 


70.66 


52754 6 


18 


43 


476498 


69.91 


805 




476693 


69.98 


523307 


17 


44 


480693 


69.24 


801 


.06 


480892 


69.31 


519108 


16 


45 


484848 


68.59 


797 


.07 


485050 


68.65 


5 J 495o 


15 


46 


488963 


67.94 


793 




489170 


68.01 


510830 


14 


47 


493040 


67.31 


79° 




493250 


67.38 


506750 


13 


48 


497078 


66.69 


786 




497 2 93 


66.76 


502707 


12 


49 


501080 


66.08 


782 




501298 


66.15 


498702 


11 


50 
51 


505045 
8.508974 


65.48 


778 




505267 
8.509200 


6 5-55 


494733 
11.490800 


10 
9 


64.89 


9-999774 


64.96 


52 


512867 


64.32 


769 




513098 


64.39 


486902 


8 


53 


516726 


63.75 


765 




516961 


63.82 


483039 


7 


54 


520551 


63.19 


761 




520790 


63.26 


479210 


6 


55 


5M343 


62.64 


757 




524586 


62.72 


4754H 


5 


56 


528102 


62.11 


753 




528349 


62.18 


471651 


4 


57 


531828 


61.58 


748 




532080 


61.65 


467920 


3 


58 


535523 


61.06 


744 




535779 


61.13 


464221 


2 


59 


539186 


60.55 


74° 


•07 


539447 


60.62 


460553 


1 


60 


8.542819 
Cosine. 


Diff. 1" 


9-999735 
Sine. 




8.543084 
Cotang. 




11.456916 
Tang. 



M. 


Diff.l" 


Diff. 1" 


1 J 


n° 








88° 



43 



2< 


i 




LOGARITHMIC 




177° 




M. 


Sine. 


Diff. 1" 


Cosine. 


Diff.l" 

.07 


Tang. 


Diff. 1" 


Cotang. 


~60 _ 







8.542819 


60.04 


9-999735 


8.543084 


60.12 


11.456916 




1 


46422 


59-55 


73 1 


.07 


46691 


59.62 


533°9 


59 




2 


49995 


59.06 


726 


.07 


50268 


59.14 


49732 


58 




3 


53539 


58.58 


722 


.08 


53 8l 7 


58.66 


46183 


57 




4 


57054 


58.11 


717 




5733 6 


58.19 


42664 


56 




5 


60540 


57-65 


713 




60828 


57-73 


39172 


55 




6 


63999 


57-19 


708 




64291 


57.27 


35709 


54 




7 


6743 1 


56.74 


704 




67727 


56.82 


32273 


53 




8 


70836 


56.30 


699 




71137 


56.38 


28863 


52 




y 


74214 


55-87 


694 




74520 


55-95 


25480 


51 




10 

n 


77566 


55-44 


689 
9.999685 




77877 


55-52 


22123 


50 
49 




8.580892 


55.02 


8.581208 


55-io 


11.418792 




12 


84193 


54.60 


680 




84514 


54.68 


15486 


48 




13 


87469 


54-19 


675 




87795 


54-27 


12205 


4JT 




14 


90721 


53-79 


670 




91051 


53-87 


08949 


46 




15 


93948 


53-39 


665 




94283 


53-47 


05717 


45 




16 


8.597152 


53-°° 


660 




8.597492 


53.08 


11.402508 


44 




17 


8.600332 


52.61 


655 




8.600677 


52.70 


11.399323 


43 




18 


03489 


52.23 


650 


.08 


03839 


52.32 


96161 


42 




19 


06623 


51.86 


645 


.09 


06978 


51.94 


93022 


41 




20 
21 


09734 


51.49 


640 




10094 


51.58 


89906 


40 
39 




8.612823 


51.12 


9-999 6 35 


8.613189 


51.21 


11.386811 




22 


15891 


50.76 


629 




16262 


50.85 


83738 


38 




23 


18937 


50.41 


624 




i93 J 3 


50.50 


80687 


37 




24 


21962 


50.06 


619 




22343 


50.15 


77657 


36 




25 


24965 


49.72 


614 




25352 


49.81 


74648 


35 




26 


27948 


49.38 


608 




28340 


49-47 


71660 


34 




27 


30911 


49.04 


603 




31308 


49-13 


68692 


33 




28 


33 8 54 


48.71 


597 




3425 6 


48.80 


65744 
62816 


32 




29 


36776 


4 8 -39 


592 




37184 


48.48 


31 




30 
31 


39680 
8.642563 


48.06 


586 




40093 


48.16 


59907 


30 
29 




47-75 


9.999581 


8.642983 


47.84 


11. 357017 




32 


45428 


47-43 


575 




45853 


47-53 


54H7 


28 




33 


48274 


47.12 


57o 




48704 


47.22 


51296 


27 




34 


51102 


46.82 


564 


.09 


5 J 537 


46.91 


48463 


26 




35 


539 11 


46.52 


558 


.10 


54352 


46.61 


45648 


25 




36 


56702 


46.22 


553 




57H9 


46.31 


42851 


24 




37 


59475 


45.92 


547 




59928 


46.02 


40072 


23 




38 


62230 


45.63 


54i 




62689 


45-73 


373" 


22 




39 


64968 


45-35 


535 




65433 


45-44 


34567 


21 




40 
41 


67689 


45.06 


529 




68160 


45.16 


31840 


20 
19 




8.670393 


44-79 


9.999524 


8.670870 


44.88 


11. 329130 




42 


73080 


44.51 


518 




73563 


44.61 


26437 


18 




43 


75751 


44.24 


512 




76239 


44-34 


23761 


17 




44 


78405 


43-97 


506 




78900 


44.07 


21 100 


16 




45 


81043 


43.70 


500 




81544 


43.80 


18456 


15 




46 


83665 


43-44 


493 




84172 


43-54 


15828 


14 




47 


86272 


43.18 


487 




86784 


43.28 


13216 


13 




48 


88863 


42.92 


481 




89381 


43-°3 


10619 


12 




49 


91438 


42.67 


475 




91963 


42.77 


08037 


11 




50 
51 


93998 
9 6 543 


42.42 


469 


.10 
.11 


94529 


42.52 


o547i 


10 
9 




42.17 


9.999463 


97081 


42.28 


02919 




52 


8.699073 


41.92 


456 




8.699617 


42.03 


11.300383 


8 




53 


8.701589 


41.68 


450 




8.702139 


41.79 


11. 297861 


7 




54 


04090 


41.44 


443 




04646 


41-55 


95354 


6 




00 


06577 


41.21 


437 




07140 


4.1.32 


92860 


5 




56 


09049 


40.97 


431 




09618 


41.08 


90382 


4 




57 


1 1 507 


40.74 


424 




12083 


40.85 


87917 


3 




58 


I395 2 


40.51 


418 




14535 


40.62 


85465 


2 




59 


16383 


40.29 


411 


.11 


16972 


40.40 


83028 


1 




60 


8.718800 




9.999404 


Diff.l" 


8.719396 
Cotang. 




11.280604 
Tang. 



M. 




Cosine. 


Diff. 1" 


Sine. 


Diff. 1" 




! 


)2° 








87° 





44 



3 


3 


SINES AND TANGENTS, 


176° 


M. 



Sine. 


DiflF. 1" 


Cosine. 


Diff.l" 
.11 


Tang. 


Diff. 1" 


Cotang. 
II.280604 


60 


8.718800 


40.06 


9.999404 


8.719396 


40.17 


1 


21204 


39- 8 4 


9398 




21806 


39-95 


78194 


59 


2 


23595 


39.62 


9391 




24203 


39-74 


75797 


58 


8 


25972 


39-41 


9384 




26588 


39-52 


73412 


57 


4 


28337 


39- J 9 


9378 




28959 


39-3i 


71041 


56 


5 


30688 


38.98 


9371 


.11 


3I3I7 


39-°9 


68683 


55 


6 


33027 


3 8 -77 


9364 


.12 


33663 


38.89 


66337 


54 


7 


35354 


3f.57 


9357 




35996 


38.68 


64004 


53 


8 


37667 


38.36 


935° 




3 8 3 J 7 


38.48 


61683 


52 


9 


39969 


38.16 


9343 




40626 


38.27 


59374 


51 


10 

11 


42259 


37.96 


9336 




42922 


38.07 


57078 


50 
49 


8.744536 


37-76 


9.999329 




8.745207 


37-87 


"•254793 


12 


46802 


37.56 


9322 




47479 


37.68 


52521 


48 


13 


49°55 


37-37 


93*5 




49740 


37-49 


50260 


47 


14 


5 I2 97 


37-17 


9308 




51989 


37-29 


48011 


46 


15 


535 28 


36.98 


9301 




54 22 7 


37.10 


45773 


45 


16 


55747 


3 6 -79 


9294 




56453 


36.92 


43547 


44 


17 


57955 


36.61 


9286 




58668 


3 6 -73 


41332 


43 


18 


60151 


36.42 


9279 




60872 


36-55 


39128 


42 


19 


62337 


36.24 


9272 




63065 


36-36 


36935 


41 


20 
21 


645 1 1 


36.06 


9265 




65246 
8.767417 


36.18 


. 34754 


40 
39 


8.766675 


35.88 


9.999257 


.12 


36.00 


11.232583 


22 


68828 


35-7° 


9250 


•13 


69578 


35-83 


30422 


38 


28 


70970 


35-53 


9242 




71727 


35-65 


28273 


37 


24 


73101 


35-35 


9235 




73866 


35-48 


26134 


36 


25 


75223 


35-i8 


9227 




75995 


35-31 


24005 


35 


26 


77333 


35-°i 


9220 




78114 


35-H 


21886 


34 


27 


79434 


34.84 


9212 




80222 


34-97 


19778 


33 


28 


81524 


34-67 


9205 




82320 


34.80 


17680 


32 1 


29 


83605 


34-5i 


9197 




84408 


34.64 


15592 


31 


30 
31 


85675 


34-34 


9189 




86486 
8.78*554 


34-47 


I35H 


30 
29 


8.787736 


34.18 


9.999181 




34-31 


11.211446 


32 


89787 


34.02 


9 J 74 




90613 


34-15 


09387 


28 


33 


91828 


33.86 


9166 




92662 


33-99 


07338 


27 


34 


93 8 59 


33-7° 


9158 




94701 


33-83 


05299 


26 


85 


95881 


33-54 


9150 




96731 


33-68 


03269 


25 


36 


97894 


33-39 


9142 




8.798752 


33-52 


11. 201248 


24 


87 


8.799897 


33-2-3 


9 J 34 




8.800763 


33-37 


11. 199237 


23 


38 


8.801892 


33.08 


9126 




02765 


33.22 


97235 


22 


39 


03876 


32-93 


9118 




04758 


33-°7 


95242 


21 


40 
41 


05852 


32.78 


9110 




06742 


32.92 


93258 
11.191283 


20 
19 


8.807819 


32.63 


9.999102 


•13 


8.808717 


32-77 


42 


09777 


3 2 -49 


9094 


.14 


10683 


32.62 


893^ 


18 


43 


11726 


3 2 -34 


9086 




12641 


32.48 


87359 


17 


44 


13667 


32.19 


9077 




14589 


32-33 


85411 


16 


45 


15599 


32.05 


9069 




16529 


32.19 


83471 


15 


46 


17522 


31.91 


9061 




1 846 1 


32.05 


8i539 


14 


47 


19436 


3 J -77 


9053 




20384 


31.91 


79616 


13 


48 


21343 


31.63 


9044 




22298 


3 x -77 


77702 


12 


49 


23240 


31.49 


9036 




24205 


31.63 


75795 


11 


50 
51 


25130 


3 J -35 
31.22 


9027 




26103 


31-5° 


73897 
11. 172008 


10 
9 


8.827011 


9.999019 




8.827992 


31.36 


52 


28884 


31.08 


9010 




29874 


31.23 


70126 


8 


53 


3°749 


3°-95 


9002 




3^48 


31.09 


68252 


7 


54 


32607 


30.82 


8993 




33613 


30.96 


66387 


6 


55 


3445 6 


30.69 


8984 




3547i 


30.83 


64529 


5 


56 


36297 


30.56 


8976 


.14 


3732i 


30.70 


62679 


4 


57 


38130 


3°-43 


8967 


•15 


39163 


3°-57 


60837 


3 


58 


39956 


3°-3° 


8958 


•15 


40998 


3°-45 


59002 


2 


59 


41774 


30.17 


8950 


■15 


42825 


3°-3 2 


57175 


1 


60 


8.843585 




9.998941 
Sine. 


Diff.l" 


8.844644 


Diff. 1" 


11. 155356 



M. 


Cosine. 


Diff. 1" 


Cotang. 


Tang. 


9 


3° 








86° 



45 





4: C 


> 




XiOaARXTHMIC 




175° 






M. 



Sine. 


Diff. 1" 1 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 






8.843585 


30.05 


9.998941 


•15 


8.844644 


30.19 


U-I5535 6 






i 


453 8 7 


29.92 


932 




46455 


30.07 


53545 


59 






2 


47183 


29.80 


923 




48260 


29.95 


5i74o 


58 | 






3 


48971 


29.67 


914 




5°°57 


29.82 


49943 


57 | 






4 


5°75i 


29.55 


905 




51846 


29.70 


48i54 


56 i 






5 


52525 


29.43 


896 




53628 


29.58 


46372 


55 






6 


54291 


29.31 


887 




55403 


29.46 


44597 


54 






V 


56049 


29.19 


878 




57i7i 


29.35 


42829 


53 






8 


57801 


29.08 


869 




58932 


29.23 


41068 


52 






9 


59546 


28.96 


860 




60686 


29.II 


393 J 4 


51 






10 
11 


61283 


28.84 


851 




62433 


29.00 


37567 


50 
49 






8.863014 


28.73 


9.998841 




8.864173 


28.88 


11. 135827 






12 


64738 


28.61 


832 


•15 


65906 


28.77 


34094 


48 






13 


66455 


28.50 


823 


.16 


67632 


28.66 


32368 


47 






14 


68165 


28.39 


813 




6935 1 


28.54 


30649 


46 






15 


69868 


28.28 


804 




71064 


28.43 


28936 


45 






16 


71565 


28.17 


795 




72770 


28.32 


27230 


44 






17 


73^55 


28.06 


785 




74469 


28.21 


25531 


43 






18 


74938 


27.95 


776 




76162 


28.11 


23838 


42 






19 


76615 


27.84 


766 




77849 


28.00 


22151 


41 






2U 
21 


78285 


27.73 


757 




79529 
8.881202 


27.89 


20471 


40 
39 






8.879949 


27.63 


9.998747 




27.79 


11.118798 






22 


81607 


27.52 


738 




82869 


27.68 


17131 


38 






23 


83258 


27.42 


728 




8453o 


27.58 


15470 


37 






24 


84903 


27.31 


718 




86185 


27.47 


13815 


36 






25 


86542 


27.21 


708 




87833 


27-37 


12167 


35 






26 


88174 


27.II 


699 




89476 


27.27 


10524 


34 






27 


89801 


27.00 


689 




91112 


27.17 


08888 


33 






28 


91421 


26.90 


679 


.16 


92742 


27.07 


07258 


32 






29 


93°35 


26.80 


669 


•17 


94366 


26.97 


05634 


31 






30 
~31 


94643 


26.70 


659 




95984 


26.87 
26.77 


04016 


30 
29 






96245 


26.60 


9.998649 




97596 


02404 






32 


97842 


26.51 


639 




8.899203 


26.67 


11. 100797 


28 






33 


8.899432 


26.41 


629 




8.900803 


26.58 


11. 099197 


27 






34 


8.901017 


26.31 


619 




02398 


26.48 


97602 


26 






35 


02596 


26.22 


609 




03987 


26.38 


96013 


25 






36 


04169 


26.12 


599 




05570 


26.29 


9443° 


24 






37 


05736 


26.03 


589 




07147 


26.20 


92853 


23 






38 


07297 


25-93 


578 




08719 


26.10 


91281 


22 






39 


08853 


25.84 


568 




10285 


26.01 


89715 


21 






40 
41 


10404 
8.911949 


25-75 
25.66 


558 




1 1 846 


25.92 


88154 
11.086599 


20 
19 






9.998548 




8.913401 


25.83 






42 


13488 


25-56 


537 




14951 


25.74 


85049 


18 






43 


15022 


25.47 


527 


•17 


16495 


25.65 


83505 


17 






44 


16550 


25-38 


516 


.18 


18034 


25.56 


81966 


16 






45 


18073 


25.29 


506 




19568 


25.47 


80432 


15 






46 


19591 


25.20 


495 




21096 


25.38 


78904 


14 






4V 


21103 


25.12 


485 




22619 


25.30 


7738i 


13 






48 


22610 


25.03 


474 




24136 


25.21 


75864 


12 






49 


24112 


24.94 


464 




25649 


25.12 


74351 


11 






50 
51 


25609 
8.927100 


24.86 

24.77 


453 




27156 
8.928658 


25.03 
24-95 


72844 
11-071342 


10 
9 






9.998442 








52 


28587 


24.69 


43i 




3oi55 


24.86 


69845 


8 






53 


30068 


24.60 


421 




3 l6 47 


24.78 


68353 


7 






54 


3*544 


24.52 


410 




33*34 


24.70 


66866 


6 






55 


33 OI 5 


24.43 


399 




34616 


24.61 


65384 


5 






56 


34481 


24-35 


388 




36093 


24-53 


63907 


4 






57 


3594 2 


24.27 


377 




37565 


24.45 


62435 


3 






58 


3739 8 


24.19 


366 




39032 


24.37 


60968 


2 






59 


38850 


24.11 


355 


.18 


40494 


24.29 


595o6 


1 






60 


8.940296 
Cosine. 




9.998344 

Sine. 


Diff.l" 


8.941952 




11.058048 



M. 






Diff. 1" 


Cotang. 


Diff. 1" 


Tang. 






9- 


4° 




■ 




85° 





46 



5 





SINES AND TANGENTS. 


174° 




M. 




Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 




8.940296 


24.03 


9.998344 


.19 


8.941952 


24.21 


II.058048 




1 


41738 


23.94 


333 




43404 


24.13 


56596 


59 




2 


43 J 74 


23.87 


322 




44852 


24-05 


55148 


58 




3 


44606 


23.79 


3 11 




46295 


23-97 


53705 


5/ 




4 


4 6o 34 


23.71 


300 




47734 


23.90 


52266 


56 




6 


4745 6 


23-63 


289 




49168 


a3.8a 


50832 


55 




6 


48874 


^3-55 


277 




5°597 


23-74 


49403 


54 




7 


50287 


23.48 


266 




52021 


33.66 


47979 


53 




8 


51696 


23.40 


255 




53441 


23-59 


4 6 559 


52 




y 


53100 


23-3 2 


243 




54856 


23-5I 


45 J 44 


51 




10 

n 


54499 
8.955894 


23.25 


232 




56267 


23-44 


43733 


50 
49 




23.17 


9.998220 




8.957674 


23-37 


11.042326 




12 


57284 


a3.r0 


209 




59°75 


33.39 


40925 


48 




13 


58670 


23.02 


197 




60473 


23.22 


395 2 7 


47 




14 


60052 


22.95 


186 




61866 


23.14 


38i34 


46 




15 


61429 


22.88 


174 




63255 


23.07 


36745 


45 




16 


62801 


22.80 


163 




64639 


23.00 


3536i 


44 




17 


64170 


22.73 


151 


.19 


66019 


22.93 


3398i 


43 




18 


65534 


22.66 


139 


.20 


67394 


22.86 


32606 


42 




19 


66893 


aa.59 


128 




68766 


22.79 


3 I2 34 


41 




20 
i 21 


68249 
8.969600 


22.52 


116 




7oi33 


22.71 


29867 


40 
39 




22.45 


9.998104 




8.971496 


22.65 


11.028504 




22 


70947 


22.38 


092 




72855 


22.57 


27145 


38 




23 


72289 


22.31 


080 




74209 


22.51 


25791 


37 




24 


73628 


22.24 


068 




7556o 


22.44 


24440 


36 




25 


74962 


22.17 


056 




76906 


22.37 


23094 


35 




26 


76293 


aa.io 


044 




78248 


22.30 


21752 


34 




27 


77619 


aa.o3 


032 




79586 


22.23 


20414 


33 




28 


78941 


21-97 


020 




80921 


22.17 


19079 


32 




29 


80259 


ai.90 


9.998008 




82251 


22.10 


17749 


31 




30 
31 


81573 


21.83 


9.997996 




83577 


22.04 


16423 


30 
29 




8.982883 


21-77 


984 




8.984899 


21.97 


11.015101 




32 


84189 


ai.70 


972 




86217 


21.91 


13783 


28 




33 


85491 


ai.63 


959 




87532 


21.84 


12468 


27 




34 


86789 


21-57 


947 


.20 


88842 


21.78 


11158 


26 




35 


88083 


ai.50 


935 


.21 


90149 


21.71 


09851 


25 




36 


8 9374 


a 1. 44 


922 




91451 


21.65 


08549 


24 




37 


90660 


21.38 


910 




92750 


21.58 


07250 


23 




38 


9 J 943 


21.31 


897 




94045 


21.52 


05955 


22 




39 


93222 


21.25 


885 




95337 


21.46 


04663 


21 




40 
41 


94497 


21.19 
21.12 


872 




96624 


21.40 


03376 


20 
19 




8.995768 


9.997860 




97908 


21.34 


02092 




42 


97036 


21.06 


847 




8.999188 


21.27 


11. 000812 


18 




43 


98299 


ai.oo 


835 




9.000465 


21.21 


io-999535 


17 




44 


8.999560 


ao.94 


822 




01738 


21.15 


98262 


16 




45 


9.000816 


ao.88 


809 




03007 


21.09 


96993 


15 




46 


02069 


ao.82 


797 




04272 


21.03 


95728 


14 




47 


03318 


20.76 


784 




05534 


20.97 


94466 


13 




48 


04563 


20.70 


771 




06792 


20.91 


93208 


12 




49 


05805 


20.64 


758 




08047 


20.85 


9*953 


11 




50 
51 


07044 


ao.58 


745 




09298 


20.80 


90702 


10 

9 




9.008278 


20.52 


9.997732 




9.010546 


20.74 


10.989454 




52 


09510 


20.46 


719 




11790 


20.68 


88210 


8 




53 


10737 


20.40 


706 


.ai 


13031 


20.62 


86969 


7 




54 


11962 


20.34 


693 


.a2 


14268 


20.56 


85732 


6 




55 


1318a 


30.29 


680 




1550a 


20.51 


84498 


5 




56 


14400 


20.23 


667 




16732 


20.45 


83268 


4 




57 


15613 


20.17 


654 




17959 


20.40 


82041 


3 




58 


16824 


20.12 


641 




19183 


20.34 


80817 


2 




) 59 


18031 


20.06 


628 


.22 


20403 


20.28 


79597 


1 




1 60 

i 


9.019235 




9.997614 




9.021620 




10.978380 



M. 




Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1" 


Tang. 




I 9. 


5° 








84° J 





47 



6 


3 




LOGARITHMIC 




j 

173° 


M. 


Sine. 


Diff. 1" 


Cosine. 


Diff.l" 
.22 


Tans. 


Diff. 1" 


Cotang. 
IO.978380 


60 





9.019235 


20.00 


9.997614 


9.021620 


20.23 


1 


20435 


19.95 


601 




22834 


20.17 


77166 


59 


2 


21632 


19.89 


588 




24044 


20.11 


75956 


58 


3 


22825 


19.84 


574 




25251 


20.06 


74749 


57 


4 


24016 


19.78 


561 




26455 


20.01 


73545 


56 


5 


25203 


19-73 


547 


.22 


27655 


19.95 


72345 


55 


6 


26386 


19.67 


534 


•23 


28852 


19.90 


71148 


54 


7 


27567 


19.62 


520 




30046 


19.85 


69954 


53 


8 


28744 


19-57 


5°7 




31237 


19.79 


68763 


52 


9 


29918 


I9-5I 


493 




3 2 4 2 5 


19.74 


67575 


51 


10 
11 


31089 


19.46 


480 
9.997466 




33609 


19.69 


66391 


50 
49 


9.032257 


19.41 




9.034791 


19.64 


10.965209 


12 


334 2 i 


19.36 


45* 




35969 


19.58 


64031 


48 


13 


345 82 


19.30 


439 




37H4 


19-53 


62856 


47 


14 


35741 


19.25 


425 




38316 


19.48 


61684 


46 


15 


36896 


19.20 


411 




39485 


19-43 


60515 


45 


16 


38048 


19.15 


397 




40651 


19.38 


59349 


44 


IV 


39197 


19.10 


383 




41813 


19-33 


58187 


43 


18 


40542 


19.05 


3 6 9 




42973 


19.28 


57027 


42 


19 


41485 


18.99 


355 




44I3 


19.23 


55870 


41 


20 
21 


42625 


18.95 


34i 


•23 


45284 


19.18 


547i6 


40 
39 


9.043762 


18.89 


9.997327 


.24 


9.046434 


19.13 


10.953566 


22 


44895 


18.84 


3*3 




47582 


19.08 


52418 


38 


23 


46026 


18.79 


299 




48727 


19.03 


51273 


37 


24 


47154 


18.75 


285 




49869 


18.98 


50131 


36 


25 


48279 


18.70 


271 




51008 


18.93 


48992 


35 


26 


49400 


18.65 


257 




52144 


18.89 


47856 


34 


27 


5 5i9 


18.60 


242 




53277 


18.84 


46723 


33 


28 


5 l6 35 


18.55 


228 




544°7 


18.79 


45593 


32 


29 


5 2 749 


18.50 


214 




55535 


18.74 


44465 


31 


30 
31 


53 8 59 


18.45 


199 




56659 


18.70 


4334 1 


30 

29 | 


9.054966 


18.41 


9.997185 




9.057781 


18.65 


10.942219 


32 


56071 


18.36 


170 




58900 


18.60 


41100 


28 


33 


57*72 


18.31 


156 




60016 


18.55 


39984 


27 


34 


58271 


18.27 


141 




61130 


18.51 


38870 


26 


35 


593 6 7 


18.22 


127 




62240 


18.46 


37760 


25 


36 


60460 


18.17 


112 




63348 


18.42 


36652 


24 


37 


61551 


18.13 


098 


.24 


64453 


18.37 


35547 


23 


38 


62639 


18.08 


083 


•25 


65556 


18.33 


34444 


22 


39 


63724 


18.04 


068 




66655 


18.28 


33345 


21 


40 
41 


64806 


17.99 


053 




67752 


18.24 


32248 


20 
19 


9.065885 


17.94 


9.997039 




9.068846 


18.19 


10.931154 


42 


66962 


17.90 


024 




69938 


18.15 


30062 


18 


43 


68036 


17.86 


9.997009 




71027 


18.10 


28973 


17 


44 


69107 


17.81 


9.996994 




72113 


18.06 


27887 


16 


45 


70176 


17.77 


979 




73 J 97 


18.02 


26803 


15 


46 


71242 


17.72 


964 




74278 


17.97 


25722 


14 


47 


72305 


17.68 


949 




75356 


17.93 


24644 


13 


48 


73366 


17.63 


934 




76432 


17.89 


23568 


12 


49 


74424 


17-59 


919 




77505 


17.84 


22495 


11 


50 
"51 


75480 
9.076533 


17-55 


904 




78576 


17.80 


21424 


10 
9 


17-5° 


9.-996889 




9.079644 


17.76 


10.920356 


52 


775 8 3 


17.46 


874 




80710 


17.72 


19290 


8 


53 


78631 


17.42. 


858 




8i773 


17.67 


18227 


7 


54 


79676 


I7-38 


843 




82833 


17.63 


17167 


6 


55 


80719 


17-33 


828 


.25 


83891 


17-59 


16109 


5 


56 


8i759 


17.29 


812 


.26 


84947 


17-55 


15053 


4 


57 


82797 


17.25 


797 




86000 


17.51 


14000 


3 


58 


83832 


17.21 


782 




87050 


17-47 


12950 


2 


59 


84864 


17.17 


766 


.26 


88098 


17-43 


11902 


1 


60 


9.085894 




9.996751 


Diff.l" 


9.089144 
Cotang. 




10.910856 
Tang. 



M. 


Cosine. 


Diff. 1" 


Sine. 


Diff. 1" 


9 


6° 








83° 



48 





7 


D 


SINES AND TANGENTS. 


172° 




M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 




9.085894 


I7-I3 


9.996751 


.26 


9.089144 


I7-38 


IO.910856 




i 


86922 


17.09 


735 




90187 


17-35 


09813 


59 




2 


8 7947 


17.04 


720 




91228 


17.30 


08772 


58 




3 


88970 


17.00 


704 




92266 


17.27 


07734 


57 




4 


89990 


16.96 


688 




93302 


17.22 


06698 


56 




5 


91008 


16.92 


673. 




94335 


17.19 


05664 


55 




6 


92024 


16.88 


657 




95367 


17.15 


04633 


54 




7 


93037 


16.84 


641 




96395 


17.11 


03605 


53 




8 


94047 


16.80 


625 




97422 


17.07 


02578 


52 




9 


95056 


16.76 


610 




98446 


17.03 


01554 


51 




10 

11 


96062 


16.73 


594 
9.996578 


.26 


99468 


16.99 


IO.900532 


50 
49 




9.097065 


16.68 


.27 


9.100487 


16.95 


IO.899513 




12 


98066 


16.65 


562 




01504 


16.91 


98496 


48 




13 


9.099065 


16.61 


546 




02519 


16.87 


97481 


47 




14 


9.100062 


16.57 


53o 




03532 


16.84 


96468 


46 




15 


01056 


16.53 


514 




04542 


16.80 


95458 


45 




16 


02048 


16.49 


498 




05550 


16.76 


94450 


44 




17 


03037 


16.45 


482 




06556 


16.72 


93444 


43 




18 


04025 


16.42 


465 




°7559 


16.69 


92441 


42 




iy 


05010 


16.38 


449 




08560 


16.65 


91440 


41 




20 
21 


°599a 

9.106973 


16.34 


433 




°9559 
9.110556 


16.61 


' 90441 


40 
39 




16.30 


9.996417 




16.58 


10.889444 




■ 22 


07951 


16.27 


400 




11551 


16.54 


88449 


38 




23 


08927 


16.23 


384 




12543 


16.50 


87457 


37 




24 


09901 


16.19 


368 




13533 


16.47 


86467 


36 




25 


10873 


16.16 


35i 




14521 


16.43 


85479 


35 




26 


1 1 842 


16.12 


335 




15507 


16.39 


84493 


34 




27 


12809 


16.08 


318 


.27 


16491 


16.36 


83509 


33 




28 


13774 


16.05 


302 


.28 


17472 


16.32 


82528 


32 




29 


14737 


16.01 


285 




18452 


16.29 


81548 


31 




30 

I- — 
31 


15698 


15-97 


269 




19429 


16.25 


80571 


30 
29 




9.116656 


15.94 


9.996252 




9.120404 


16.22 


10.879596 




! 32 


17613 


15.90 


235 




21377 


16.18 


78623 


28 




1 33 


18567 


15.87 


219 




22348 


16.15 


77652 


27 




i 34 


19519 


15.83 


202 




23317 


16.11 


76683 


26 




35 


20469 


15.80 


185 




24284 


16.08 


75716 


25 




36 


21417 


15.76 


168 




25249 


16.04 


74751 


24 




37 


22362 


15-73 


151 




26211 


16.01 


73789 


23 




38 


23306 


15.69 


134 




27172 


15-97 


72828 


22 




39 


24248 


15.66 


117 




28130 


15.94 


71870 


21 




40 
41 


25187 


15.62 


100 


.28 


29087 


15.91 


70913 


20 
19 




9.126125 


15-59 


9.996083 


.29 


9.130041 


15.87 


10.869959 




42 


27060 


15.56 


066 




3°994 


15.84 


69006 


18 




43 


27993 


15.52 


049 




3*944 


15.81 


68056 


17 




44 


28925 


15.49 


032 




32893 


15-77 


67107 


16 




45 


29854 


15-45 


9.996015 




33 8 39 


15-74 


66161 


15 




46 


30781 


15.42 


9.995998 




34784 


I5-7I 


65216 


14 




47 


31706 


15-39 


980 




35726 


15.67 


64274 


13 




48 


32630 


15-35 


963 




36667 


15.64 


63333 


12 




49 


33551 


15.32 


946 




37605 


15.61 


62395 


11 




: 50 
i 51 , 


3447° 
9- I 353 8 7 


15.29 


928 




3 8 542 


15.58 


61458 
10.860524 


10 
9 




15-^5 


9.995911 




9.139476 


15-55 




o2 


36303 


15.22 


894 




40409 


15.51 


5959 1 


8 




53 


37216 


I5-I9 


876 




4 X 34° 


15.48 


58660 


7 




54 


38128 


15.16 


859 




42269 


15-45 


5773 1 


6 




1 55 


39037 


15.12 


841 




43196 


15.42 


56804 


5 




56 


39944 


15.09 


823 




441 2 1 


'5-39 


55879 


4 




57 


40850 


15.06 


806 




45°44 


15-35 


54956 


3 




58 


41754 


i5- 3 


788 




45966 


I5-3 2 


54034 


2 




59 


42655 


15.00 


771 


.29 


46885 


15.29 


53II5 


1 




60 


9-H3555 
Cosine. 




9-995753 




9.147803 




10.852197 



M. 




Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1" 


Tang. 




9 


7° 








82° 



49 



8 


J 




LOGARITHMIC 


171° | 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 
9.147803 


Diff. 1" 


Cotang. 


60 | 


9-*43555 


14.96 


9-995753 


.30 


15.26 


IO.852197 


1 


4453 


14.93 


735 




8718 


i5- 2 3 


1282 


59 j 


2 


5349 


14.90 


717 




9.149632 


15.20 


IO.850368 


58 1 


8 


6243 


14.87 


699 




9.150544 


15.17 


10.849456 


57 | 


4 


7136 


14.84 


681 




1454 


15.14 


8546 


56 


5 


8026 


14.81 


664 




2363 


15.11 


7637 


55 


6 


8915 


14.78 


646 




3269 


15.08 


6731 


54 


7 


9.149802 


14-75 


628 




4174 


i5-o5 


5826 


53 


8 


9.150686 


14.72 


610 




5°77 


15.02 


4923 


52 


y 


1569 


14.69 


59i 




5978 


14.99 


4022 


51 


10 

n 


2451 


14.66 


573 




6877 


14.96 


3123 


50 

49 


9- I 5333° 


14.63 


9-995555 




9- J 57775 


14.93 


10.842225 


12 


4208 


14.60 


537 




8671 


14.90 


1329 


48 


13 


5083 


14-57 


519 


.30 


9- J 595 6 5 


14.87 


10.840435 


47 


14 


5957 


14-54 


501 


•31 


9.160457 


14.84 


IO.839543 


46 


lb 


6830 


H-5i 


482 




1347 


14.81 


8653 


45 


16 


7700 


14.48 


464 




2236 


14.78 


7764 


44 


17 


8569 


14.45 


446 




3123 


14-75 


6877 


43 


18 


9-159435 


14.42 


427 




4008 


14-73 


599 2 


42 


iy 


9.160301 


14-39 


409 




4892 


14.70 


5108 


41 


20 
21 


1 1 64 


14.36 


390 




5774 

9.166654 


14.67 


4226 


40 
39 


9.162025 


14-33 


9-99537* 




14.64 


10.833346 


22 


2885 


14.30 


353 




753 2 


14.61 


2468 


38 


23 


3743 


14.27 


334 




8409 


14.58 


1591 


37 


24 


4600 


14.24 


316 




9.169284 


14-55 


10.830716 


36 


2o 


5454 


14.22 


297 




9.170157 


14-53 


10.829843 


35 


26 


6307 


14.19 


278 




1029 


14.50 


8971 


34 


27 


7i59 


14.16 


260 


•31 


1899 


14.47 


8101 


33 


28 


8008 


14.13 


241 


.32 


2767 


14.44 


72-33 


32 


2y 


8856 


14.10 


222 




3 6 34 


14.42 


6366 


31 


30 
31 


9.169702 


14.07 


203 




4499 


14-39 


55oi 
10.824638 


30 
29 


9- J 7°547 


14.05 


9.995184 




9.175362 


14.36 


32 


1389 


14.02 


165 




6224 


14-33 


3776 


28 


33 


2230 


13.99 


146 




7084 


H-3 1 


2916 


27 


34 


3070 


13.96 


127 




7942 


14.28 


2058 


26 


35 


3908 


13-94 


108 




8799 


14.25 


1201 


25 


36 


4744 


13.91 


089 




9.179655 


14.23 


10.820345 


24 


87 


5578 


13.88 


070 




9.180508 


14.20 


10.819492 


23 


38 


6411 


13.86 


051 




1360 


14.17 


8640 


22 


3y 


7242 


13.83 


032 




2211 


14.15 


7789 


21 


40 
41 


8072 


13.80 


9.995013 




3°59 


14.12 


6941 
10.816093 


20 
19 


8900 


13-77 


9.994993 


9.183907 


14.09 


42 


9.179726 


13-74 


974 




475* 


14.07 


5248 


18 


43 


9.180551 


13.72 


955 




5597 


14.04 


44°3 


17 


44 


1374 


13.69 


935 


.32 


6 439 


14.02 


356i 


16 


45 


2196 


13.67 


916 


•33 


7280 


13-99 


2720 


15 


46 


3016 


13.64 


896 




8120 


13.96 


1880 


14 


47 


3834 


13.61 


877 




8958 


13-94 


1042 


13 


48 


4651 


13-59 


857 




9.189794 


I3-9I 


10.810206 


12 


4y 


5466 


I3-5 6 


838 




9.190629 


13.89 


10.809371 


11 


50 


6280 


13-53 


818 




1462 


13.86 
13.84 


8538 
10.807706 


10 
9 


51 


9.187092 


I3-5 1 


9.994798 




9.192294 


52 


79°3 


13.48 


• 779 




3 I2 4 


13.81 


6876 


8 


53 


8712 


13.46 


759 




3953 


13-79 


6047 


7 


54 


9.189519 


13-43 


739 




4780 


13.76 


5220 


6 


55 


9.190325 


13.41 


719 




5606 


13-74 


4394 


5 


56 


1130 


I3-38 


700 




6430 


13.71 


3570 


4 1 


57 


1933 


13.36 


68c 




7253 


13.69 


2747 


3 


58 


2734 


13-33 


660 




8074 


13.66 


1926 


2 


59 


3534 


13.30 


640 


•33 


8894 


13.64 


1106 


1 


60 


9.194332 




9.994620 




9.199713 




10.800287 

Tang. 



M. 


Cosine. 


Diff. 1" 


Sine. 


Diff.]" 


Cotang. 


Diff. 1" 




98° 






81° 














50 



i 
9° 


SXCTES AND TANaENTS. 


170° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. j 


60 


9.194332 


13.28 


9.994620 


•33 


9-I997I3 


13.61 


10.800287I 


1 


5129 


13.26 


600 


•33 


9.200529 


13-59 


10.799471 


59 


2 


59^5 


13.23 


580 


•33 


1345 


I3-5 6 


8655 


58 


3 


6719 


13.21 


560 


•34 


2159 


13-54 


7841 


57 


4 


75" 


13.18 


540 




2971 


I3-52 


7029 


56 


6 


8302 


13.16 


5 J 9 




3782 


13-49 


6218 


55 


6 


9091 


I3-I3 


499 




4592 


r 3-47 


5408 


54 


. V 


9.199879 


13.11 


479 




5400 


13-45 


4600 


53 


8 


9.200666 


13.08 


459 




6207 


13.42 


3793 


52 


y 


1451 


13.06 


438 




7013 


13.40 


2987 


51 


1U 

n 


2234 


13.04 


418 




7817 


13.38 


2183 


50 
49 


9.203017 


13.01 


9-994397 




8619 


13-35 


1 38 1 


12 


3797 


12.99 


377 




9.209420 


13-33 


10.790580 


48 


13 


4577 


12.96 


357 




9.210220 


I 3-3 I 


10.789780 


47 


14 


5354 


12.94 


33 6 




1018 


13.28 


8982 


46 


15 


6131 


12.92 


316 




1815 


13.26 


8185 


45 


16 


6906 


12.89 


295 


•34 


2611 


13.24 


7389 


44 


17 


7679 


12.87 


274 


•35 


34°5 


13.21 


6595 


43 


18 


8452 


12.85 


254 




4198 


I3-I9 


5802 


42 


19 


9222 


12.82 


233 




4989 


I3-J7 


5011 


41 


20 
21 


9.209992 


12.80 


212 




5780 


i3-!5 


4220 
10.783432 


40 
39 


9.210760 


12.78 


9.994191 




9.216568 


13.12 


22 


1526 


12.75 


171 




7356 


13.10 


2644 


38 


28 


2291 


12.73 


150 




8142 


13.08 


1858 


37 


24 


3°55 


12.71 


129 




8926 


i3-°5 


1074 


36 


25 


3818 


12.68 


108 




9.219710 


13.03 


10.780290 


35 


26 


4579 


12.66 


087 




9.220492 


13.01 


10.779508 


34 


27 


5338 


12.64 


066 




1272 


12.99 


8728 


33 


28 


6097 


12.61 


045 




2052 


12.97 


7948 


32 


29 


6854 


12.59 


024 




2830 


12.94 


7170 


31 


30 
31 


7609 
9.218363 


12.57 


9.994003 




3606 


12.92 


6394 


30 
29 


12.55 


9.993981 




9.224382 


12.90 


10.775618 


32 


9116 


i 2 -53 


960 




5i5 6 


12.88 


4844 


28 


33 


9.219868 


12.50 


939 




5929 


12.86 


4071 


27 


34 


9.220618 


12.48 


918 


•35 


6700 


12.84 


3300 


26 


35 


1367 


12.46 


896 


.36 


747i 


12.81 


2529 


25 


36 


2115 


12.44 


875 




8239 


12.79 


1761 


24 


37 


2861 


12.42 


854 




9007 


12.77 


0993 


23 


38 


3606 


12.39 


832 




9.229773 


12.75 


10.770227 


22 


39 


4349 


12.37 


811 




9.230539 


12.73 


10.769461 


21 


40 
41 


5092 


is-35 

12.33 


789 




1302 


12.71 


8698 
10.767935 


20 


9.225833 


9.993768 


9.232065 


12.69 


42 


6573 


12.31 


746 




2826 


12.67 


7174 


18 


43 


7311 


12.28 


725 




3586 


12.65 


6414 


17 


44 


8048 


12.26 


703 




4345 


12.62 


5 6 55 


16 


45 


8784 


12.24 


681 




Sio3 


12.60 


4897 


15 


46 


9.229518 


12.22 


660 




5859 


12.58 


4141 


14 


47 


9.230252 


12.20 


638 




6614 


12.56 


3386 


13 


48 


0984 


12.18 


616 


•36 


7368 


12.54 


2632 


12 


49 


1714 


12.16 


594 


•37 


8120 


12.52 


1880 


11 


50 
_ 51 


2444 


12.14 


572 




8872 


12.50 


1128 

10.760378 


10 

~~ 9~ 


9.233172 


12.12 


9-99355° 




9.239622 


12.48 


52 


3 8 99 


12.09 


528 




9.240371 


12.46 


10.759629 


8 


63 


4625 


12.07 


506 




1118 


12.44 


8882 


7 


54 


5349 


12.05 


484 




1865 


12.42 


8135 


6 


5o 


6073 


12.03 


462 




2610 


12.40 


7390 


5 


56 


6795 


12.01 


440 




3354 


12.38 


6646 


4 


57 


7515 


11.99 


418 




4097 


12.36 


59°3 


3 


58 


8235 


11.97 


396 




4839 


12.34 


5161 


2 1 


59 


8953 


11.95 


374 


•37 


5579 


12.32 


4421 


1 1 


60 


9.239670 




9-993351 




9.246319 




10.753681 

Tang. 



M. 


Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1" 


99° 










80° i 



51 



1 

10° 




LOOARXTHXftlC 




169° 




M. 


Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang:. 


Diff. 1" 


Cotang. 


60 







9.239670 


"•93 


9-993351 


•37 


9.246319 


12.30 


IO.753681 




1 


9.240386 


H.91 


3 2 9 




7057 


12.28 


2 943 


59 ! 




2 


IIOI 


II.89 


3°7 




7794 


12.26 


2206 


58 ■ 




3 


1814 


II.87 


285 




8530 


12.24 


1470 


57 




4 


2526 


II.85 


262 




9264 


12.22 


0736 


56 




5 


3*37 


II.83 


240 


•37 


9.249998 


I2.20 


10.750002 


55 




6 


3947 


II.81 


217 


•38 


9.250730 


I2.I8 


IO.749270 


54 




7 


4656 


II.79 


195 




1461 


12.17 


8539 


53 




8 


53 6 3 


II.77 


172 




2191 


12.15 


7809 


52 




y 


6069 


II.75 


149 




2920 


12.13 


7080 


51 




10 

n 


6775 


"•73 


127 




3648 


12. II 


6352 


50 
49 




9.247478 


11.71 


9.993104 




9- 2 54374 


I2.O9 


IO.745626 




12 


8181 


11.69 


081 




5100 


I2.O7 


4900 


48 




13 


8883 


11.67 


059 




5824 


I2.O5 


4176 


47 




14 


9.249583 


11.65 


036 




6547 


I2.03 


3453 


46 




15 


9.250282 


11.63 


9.993013 




7269 


I2.0I 


2731 


45 




16 


0980 


11.61 


9.992990 




7990 


I2.00 


2010 


44 




17 


1677 


11.59 


967 




8710 


II.98 


1290 


43 




18 


2373 


11.58 


944 




9.259429 


II.96 


10.740571 


42 




19 


3067 


11.56 


921 




9.260146 


II.94 


10.739854 


41 




20 
21 


3761 


11.54 


898 




0863 


II.92 


9*37 


40 
39 




9.254453 


11.52 


9.992875 




9.261578 


II.90 


10.738422 




22 


5 J 44 


11.50 


852 


.38 


2292 


II.89 


7708 


38 




2'6 


5834 


11.48 


829 


•39 


3005 


II.87 


6995 


37 




24 


6523 


11.46 


806 




3717 


II.85 


6283 


36 




25 


7211 


11.44 


783 




4428 


II.83 


5572 


35 




26 


7898 


11.42 


759 




5138 


II.8I 


4862 


34 




27 


8583 


11.41 


736 




5847 


11.79 


4153 


33 




28 


9268 


"•39 


713 




6555 


II.78 


3445 


32 




29 


9.259951 


n-37 


690 




7261 


II.76 


2739 


31 




30 
31 


9.260633 


n-35 


666 




7967 


II.74 


2033 


30 
29 




I3H 


n-33 


9.992643 




8671 


II.72 


1329 




32 


1994 


11. 31 


619 




9.269375 


II.70 


10.730625 


28 




33 


2673 


11.30 


596 




9.270077 


II.69 


10.729923 


27 




34 


335 1 


11.28 


572 




0779 


II.67 


9221 


26 




35 


4027 


11.26 


549 




1479 


II.65 


8521 


25 




36 


4703 


11.24 


525 




2178 


II.64 


7822 


24 




37 


5377 


11.22 


501 


•39 


2876 


11.62 


7124 


23 




38 


6051 


11.20 


478 


.40 


3573 


II.60 


6427 


22 




39 


6723 


11. 19 


454 




4269 


II.58 


573 1 


21 




40 
41 


7395 


11. 17 


430 




4964 


11.57 


5036 


20 
19 




9.268065 


11. 15 


9.992406 




9.275658 


11.55 


10.724342 




42 


8734 


11. 13 


382 




6351 


"•53 


3649 


18 




43 


9.269402 


11. 12 


359 




7043 


11. 51 


2957 


17 




44 


9.270069 


11. 10 


335 




7734 


11.50 


2266 


16 




45 


0735 


11.08 


3" 




8424 


11.48 


1576 


15 




46 


1400 


11.06 


287 




9"3 


11.46 


0887 


14 




47 


2064 


11.05 


263 




9.279801 


11.45 


10.720199 


13 




48 


2726 


11.03 


239 




9.280488 


11.43 


10.719512 


12 




49 


3388 


11. 01 


214 




1174 


11.41 


8826 


11 




50 
51 


4049 


10.99 


190 
9.992166 




1858 


11.40 


8142 


10 
9 




9.274708 


10.98 




9.282542 


11.38 


10.717458 




52 


5367 


10.96 


142 


.40 


3225 


11.36 


6775 


8 




53 


6024 


10.94 


117 


.41 


39°7 


11.35 


6093 


7 




54 


6681 


10.92 


°93 




4588 


11.33 


5412 


6 




55 


7337 


10.91 


069 




5268 


11.31 


473* 


5 




56 


7991 


10.89 


044 




5947 


11.30 


4°53 


4 




57 


8644 


10.87 


9.992020 




6624 


11.28 


3376 


3 




58 


9297 


10.86 


9.991996 




7301 


11.26 


2699 


2 




59 


9.279948 


10.84 


971 


.41 


7977 


11.25 


2023 


1 




60 


9.280599 




9.991947 




9.288652 




10.711348 




M. 




Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1" 


Tang. 




lioo° 








79° 





52 



11° 


SINES AND TANGENTS. 


■■,.,...—. — , 

168° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diffi,l" 


Tang. 


Diff. 1" 


Cotang. 


60 1 


9.280599 


IO.82 


9.991947 


.41 


9.288652 


II.23 


10.711348 


J 


1248 


10.81 


922 




9326 


11.22 


0674 


59 j 


2 


1897 


IO.79 


897 




9.289999 


II.20 


10.710001 


58 1 


8 


2544 


IO.77 


873 




9.290671 


II. l8 


10.709329 


57 


4 


3190 


IO.76 


848 




1342 


II. 17 


8658 


56 


5 


3^6 


IO.74 


823 




2013 


II. 15 


7987 


55 


6 


4480 


IO.72 


799 


.41 


2682 


II. 14 


7318 


54 


7 


5124 


IO.71 


774 


.42 


335° 


II. 12 


6650 


53 


8 


5766 


IO.69 


749 




4017 


II. II 


5983 


52 


9 


6408 


IO.67 


724 




4684 


II.09 


53 l6 


51 


10 
11 


7048 


10.66 


699 




5349 


II.07 


4651 


50 
49 


9.287687 


IO.64 


9.991674 




9.296013 


II.06 


10.703987 


12 


8326 


IO.63 


649 




6677 


II.04 


3323 


48 


13 


8964 


IO.61 


624 




7339 


II.03 


2661 


47 


14 


9.289600 


IO.59 


599 




8001 


II.OI 


1999 


46 


15 


9.290236 


IO.58 


574 




8662 


II.OO 


1338 


45 


16 


0870 


IO.56 


549 




9322 


IO.98 


0678 


44 


IV 


1504 


IO.54 


5 2 4 




9.299980 


IO.96 


10.700020 


43 


18 


2137 


IO.53 


498 




9.300638 


IO.95 


10.699362 


42 


19 


2768 


IO.51 


473 




1295 


IO.93 


8705 


41 


2U 
21 


3399 
9.294029 


IO.50 


448 




1951 


IO.92 


8049 


40 

~3<r 


10.48 


9.991422 




9.302607 


IO.9O 


10.697393 


22 


4658 


IO.46 


397 


.42 


3261 


IO.89 


6739 


38 


23 


5286 


IO.45 


37 2 


•43 


3914 


IO.87 


6086 


37 


24 


59 J 3 


IO.43 


346 




4567 


IO.86 


5433 


36 


25 


6 539 


IO.42 


321 




5218 


IO.84 


4782 


35 


26 


7164 


IO.40 


295 




5869 


IO.83 


4131 


34 


27 


7788 


IO.39 


270 




6519 


IO.81 


3481 


33 


28 


8412 


IO.37 


244 




7168 


IO.80 


2832 


32 


29 


9°34 


IO.36 


218 




7815 


IO.78 


2185 


31 


30 
31 


9.299655 
9.300276 


IO.34 


193 




8463 


IO.77 


1537 


30 
29 


IO.32 


9.991167 


9109 


IO.75 


0891 


82 


0895 


IO.31 


141 




9-3°9754 


IO.74 


10.690246 


28 


33 


1514 


IO.29 


"5 




9.310398 


IO.73 


10.689602 


27 


34 


2132 


IO.28 


090 




1042 


IO.7I 


8958 


26 


35 


2748 


IO.26 


064 




1685 


IO.7O 


8315 


25 


36 


3364 


IO.25 


038 




2327 


IO.68 


7673 


24 


37 


3979 


IO.23 


9.991012 




2967 


IO.67 


7033 


23 


38 


4593 


IO.22 


9.990986 




3608 


IO.65 


6392 


22 


89 


5207 


IO.20 


960 


•43 


4247 


IO.64 


5753 


21 


40 
41 


5819 


IO.19 


934 


•44 


4885 


IO.62 


5"5 


20 
~19~ 


9.306430 


10.17 


9.990908 


9.315523 


IO.61 


10.684477 


42 


7041 


IO.16 


882 




6159 


IO.60 


3841 


18 


43 


7650 


IO.14 


855 




6795 


IO.58 


3205 


17 


44 


8259 


IO.13 


829 




7430 


IO.57 


*57° 


16 


46 


8867 


IO.II 


803 




8064 


IO.55 


1936 


15 


46 


9.309474 


10.10 


777 




8697 


IO.54 


1303 


14 


47 


9.310080 


10.08 


750 




9329 


IO.53 


0671 


13 


48 


0685 


10.07 


724 




9.319961 


IO.5I 


10.680039 


12 


49 


1289 


10.06 


697 




9.320592 


IO.5O 


10.679408 


11 


50 
^51 


1893 


10.04 


671 
9.99064A 




1222 


IO.48 


8778 


10 
9 


9.312495 


10.03 


9.321851 


IO.47 


10.678149 


52 


3°97 


10.01 




2479 


IO.45 


7521 


8 


53 


3698 


10.00 


59i 




3106 


IO.44 


6894 


7 


54 


4297 


9.98 


565 




3733 


IO.43 


6267 


6 


55 


4897 


9-97 


538 


•44 


4358 


IO.4I 


5642 


* 


56 


5495 


9.96 


5 11 


•45 


4983 


IO.4O 


5017 


4 


67 


6092 


9.94 


485 




5607 


IO.39 


4393 


3 


j 58 


6689 


9-93 


458 




6231 


IO.37 


3769 


2 


59 


7284 


9.91 


43i 


•45 


6853 


IO.36 


3H7 


1 


60 


9.317879 
Cosine. 




9.990404 

Sine. 


Diff.l" 


9.327475 




10.672525 




M. 


Diff. 1" 


Cotang. 


Diff. 1" 


Tang. 


101° 










78° 



25 



53 



12° 




LOGARITHMIC 




167° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff. 1" 
•45 


Tang. 


Diff. 1" 


Cotang. 


60 


9.317879 


9.90 


9.990404 


9.327474 


IO-35 


IO.672526 


1 


8473 


9.88 


378 




8095 


io-33 


1905 


59 


2 


9066 


9.87 


351 




8715 


10.32 


1285 


58 


3 


9.319658 


9.86 


3*4 




9334 


10.30 


0666 


57 


4 


9.320249 


9.84 


297 




9-3*9953 


10.29 


IO.670047 


56 


5 


0840 


9-83 


270 




9.330570 


10.28 


IO.669430 


55 


6 


I430 


9.82 


*43 




1187 


10.26 


8813 


54 


7 


2019 


9.80 


215 




1803 


10.25 


8197 


53 


8 


2607 


9-79 


188 




2418 


10.24 


7582 


52 


9 


3*94 


9-77 


161 




3°33 


10.23 


6967 


51 


10 
11 


3780 


9.76 


134 


•45 

.46 


3646 


10.21 


6 354 


50 


9.324366 


9-75 


9.990107 


9-334*59 


10.20 


10.665741 


49 


12 


495o 


9-73 


079 




4871 


10.19 


5129 


48 


13 


5534 


9.72 


052 




5482 


10.17 


4518 


47 


14 


6117 


9.70 


9.990025 




6093 


10.16 


39°7 


46 


15 


6700 


9.69 


9.989997 




6702 


10.15 


3298 


45 


16 


7281 


9.68 


970 




7311 


10.13 


2689 


44 


17 


7862 


9.66 


942 




7919 


10.12 


2081 


43 


18 


8442 


9- 6 5 


9*5 




8527 


IO.II 


1473 


42 


19 


9021 


9.64 


887 




9 J 33 


IO.IO 


0867 


41 


20 

21 


9-3*9599 
9.330176 


9.62 


860 




9-339739 
9-34°344 


10.08 


10.660261 


40 
39 


9.61 


9.989832 




10.07 


10.659656 


22 


0753 


9.60 


804 




0948 


10.06 


9052 


38 


23 


1329 


9.58 


777 


.46 


1552 


10.04 


8448 


37 


24 


1903 


9-57 


749 


•47 


2155 


10.03 


7845 


36 


25 


2478 


9.56 


721 




*757 


10.02 


7*43 


35 


26 


3°5i 


9-54 


6 93 




3358 


IO.OI 


6642 


34 


27 


3624 


9-53 


665 




3958 


9.99 


6042 


33 


28 


4i95 


9.52 


637 




4558 


9.98 


544* 


32 


29 


4766 


9.50 


609 




5157 


9-97 


4843 


31 


30 
31 


5337 


9.49 


582 




5755 


9.96 


4*45 


30 
29 


9.335906 


9.48 


9-9 8 9553 




9-34 6 353 


9.94 


10.653647 


32 


6 475 


9.46 


5*5 




6949 


9-93 


3051 


28 


33 


7°43 


9-45 


497 




7545 


9.92 


*455 


27 


34 


7610 


9.44 


469 




8141 


9.91 


1859 


26 


35 


8176 


9-43 


441 




8735 


9.90 


1265 


25 


36 


8742 


9.41 


4i3 




93*9 


9.88 


0671 


24 


37 


9306 


9.40 


384 




9.349922 


9.87 


10.650078 


23 


38 


9.339871 


9-39 


35 6 




9-35°5H 


9.86 


10.649486 


22 


39 


9.340434 


9-37 


328 




1106 


9.85 


8894 


21 


40 
41 


0996 
9-34I558 


9.36 
9-35 


300 




1697 


9.83 


8303 


20 
19 


9.989271 




9.352287 


9.82 


10.647713 


42 


2119 


9-34 


*43 




2876 


9.81 


7124 


18 


43 


2679 


9-3* 


214 




34 6 5 


9.80 


6 535 


17 


44 


3*39 


9-3i 


186 




4°53 


9-79 


5947 


16 


45 


3797 


9-3° 


157 


•47 


4640 


9-77 


5360 


15 


46 


4355 


9.29 


128 


.48 


5227 


9.76 


4773 


14 


47 


4912 


9.27 


100 




5813 


9-75 


4187 


13 


48 


5469 


9.26 


071 




6398 


9-74 


3602 


12 


49 


6024 


9-*5 


042 




6982 


9-73 


3018 


11 


50 
51 


6579 


9.24 


9.989014 




7566 


9.71 


2434 


10 
9 


9.347134 


9.22 


9.988985 


9.358149 


9.70 


10.641851 


52 


7687 


9.21 


956 




8731 


9.69 


1269 


8 


53 


8240 


9.20 


927 




9313 


9.68 


0687 


7 


54 


8792 


9.19 


898 




9-359893 


9.67 


10.640107 


6 


55 


9343 


9.17 


869 




9.360474 


9.66 


10.639526 


5 


56 


9-349 8 93 


9.16 


840 


.48 


i°53 


9.65 


8947 


4 


57 


9-35°443 


9.15 


811 


•49 


1632 


9- 6 3 


8368 


3 


58 


0992 


9.14 


782 


•49 


2210 


9.62 


7790 


2 


59 


1540 


9.13 


753 


•49 


2787 


9.61 


7213 


1 


60 


9.352088 




9.988724 

Sine. 


Diff.1" 


9.363364 

Cotang. 


Diff. 1" 


10.636636 
Tang. 




M. 


Cosine. 


Diff. 1" 


102° 










77° 



54 



13° 


SINES AND TANGENTS. 


166° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.352088 


9.11 


9.988724 


•49 


9.363364 


9.60 


IO.636636 


1 


2635 


9.10 


8695 




3940 


9-59 


6060 


59 


2 


3181 


9.09 


8666 




45 J 5 


9.58 


5485 


58 


8 


3726 


9.08 


8636 




5090 


9-57 


4910 


57 


4 


4271 


9.07 


8607 




5664 


9-55 


4336 


56 


5 


4815 


9.05 


8578 




6237 


9-54 


3763 


55 


6 


5358 


9.04 


8548 




6810 


9-53 


3190 


54 


V 


5901 


9.03 


8519 




7382 


9.52 


2618 


53 


8 


6443 


9.02 


8489 




7953 


9.51 


2047 


52 


y 


6984 


9.01 


8460 




8524 


9-5° 


1476 


51 


10 

n 


75 2 4 


8.99 


8430 




9094 
9.369663 


9.49 


0906 


50 
49 


9.358064 


8.98 


9.988401 




9.48 


IO.630337 


12 


8603 


8.97 


8371 




9.370232 


9.46 


IO.629768 


48 


in 


9141 


8.96 


8342 


•49 


0799 


9-45 


9201 


47 


14 


9.359678 


8.95 


8312 


.50 


1367 


9.44 


8633 


46 


15 


9.360215 


8-93 


8282 




1933 


9-43 


8067 


45 


16 


0752 


8.92 


8252 




2499 


9.42 


7501 


44 


17 


1287 


8.91 


8223 




3064 


9.41 


6936 


43 


18 


1822 


8.90 


8193 




3629 


9.40 


6371 


42 


19 


2356 


8.89 


8163 




4193 


9-39 


58o7 


41 


20 
21 


2889 


8.88 


8i33 




475 6 


9.38 


5244 


40 
39 


9.363422 


8.87 


9.988103 




9-3753 I 9 


9-37 


IO.624681 


22 


3954 


8.85 


8073 




5881 


9-35 


4119 


38 


23 


4485 


8.84 


8043 




6442 


9-34 


3558 


37 


24 


5016 


8.83 


8013 




7003 


9-33 


2997 


36 


25 


5546 


8.82 


7983 




75 6 3 


9.32 


2 437 


35 


26 


6075 


8.81 


7953 




8122 


9.31 


1878 


34 


27 


6604 


8.80 


7922 




8681 


9.30 


I319 


33 


28 


7I3 1 


8.79 


7892 




9 2 39 


9.29 


0761 


32 


29 


7659 


8.78 


7862 


.50 


9797 


9.28 


0203 


31 


30 
31 


8185 


8.76 
8.75 


7832 


•5i 


9-3 8o 354 
9.380910 


9.27 


IO.619646 


30 
29 


9.368711 


9.987801 




9.26 


IO.619090 


32 


9236 


8.74 


7771 




1466 


9.25 


8534 


28 


33 


9.369761 


8-73 


7740 




2020 


9.24 


7980 


27 


84 


9.370285 


8.72 


7710 




2575 


9- 2 3 


74*5 


26 


35 


0808 


8.71 


7679 




3129 


9.22 


6871 


25 


36 


1330 


8.70 


7649 




3682 


9.21 


6318 


24 


37 


1852 


8.69 


7618 




4 2 34 


9.20 


5766 


23 


38 


2373 


8.67 


7588 




4786 


9.19 


5214 


22 


39 


2894 


8.66 


7557 




5337 


9.18 


4663 


21 


40 
41 


3414 


8.65 


7526. 
9.987496 




5888 


9.17 


4112 
10.613562 


20 
19 


9-373933 


8.64 




9.386438 


9.15 


42 


4452 


8.63 


7465 




6987 


9.14 


3 OI 3 


18 


43 


4970 


8.62 


7434 


•5i 


7536 


9- J 3 


2464 


17 


44 


5487 


8.61 


7403 


.52 


8084 


9.12 


1916 


16 


45 


6003 


8.60 


7372 




8631 


9.11 


1369 


15 


46 


6519 


8.59 


7341 




9178 


9.10 


0822 


14 


47 


7035 


8.58 


7310 




9.389724 


9.09 


10.610276 


13 


48 


7549 


8.57 


7279 




9.390270 


9.08 


10.609730 


12 


49 


8063 


8.56 


7248 




0815 


9.07 


9185 


11 


50 
51 


8577 


8.54 


7217 




1360 


9.06 
9-°5 


8640 
10.608097 


10 
~9~ 


9089 


8.53 


9.987186 


9.391903 


52 


9.379601 


8.52 


7155 




2447 


9.04 


7553 


8 


53 


9.380113 


8.51 


7124 




2989 


9.03 


701 1 


7 


54 


0624 


8.50 


7092 




353 1 


9.02 


6469 


6 


55 


"34 


8.49 


7061 




4073 


9.01 


59 2 7 


5 


56 


1643 


8.48 


7030 




4614 


9.00 


5386 


4 


57 


2152 


8.47 


6998 




5 J 54 


8.99 


4846 


3 


58 


2661 


8.46 


6967 




5694 


8.98 


4306 


2 


59 


3168 


8.45 


6936 


.52 


6233 


8.97 


3767 


1 


60 


9.383675 




9.986904 
Sine. 




9.396771 




10.603229 
Tang. 




M. 


Cosine. 


Diff. 1" 


Diff.l" 


Cotang. 


Diff. 1" 


103° 








76° 



55 



14° 




LOGARITHMIC 




165° 




M. 



Sine. 


Diff. 1" | 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 




9.383675 


8.44 


9.986904 


.52 


9.396771 


8.96 


IO.603229 




1 


4182 


8.43 


6873 


•53 


73°9 


8.96 


2691 


59 




2 


4687 


8.42 


6841 




7846 


8.95 


2154 


58 




3 


5192 


8.41 


6809 




8383 


8.94 


1617 


57 




4 


5697 


8.40 


6778 




8919 


8-93 


1081 


56 




5 


6201 


8-39 


6746 




9455 


8.92 


°545 


55 




6 


6704 


8.38 


6714 




9.399990 


8.91 


10.600010 


54 




7 


7207 


8-37 


6683 




9.400524 


8.90 


10.599476 


53 




8 


7709 


8.36 


6651 




1058 


8.89 


8942 


52 




9 


8210 


8-35 


6619 




1591 


8.88 


8409 


51 




10 
11 


8711 


8-34 


6587 




2124 


8.87 


7876 


50 
49 




9211 


8-33 


9.986555 




9.402656 


8.86 


iQ-597344 




12 


9.389711 


8.32 


6523 




3187 


8.85 


6813 


48 




13 


9.390210 


8.31 


6491 




37i8 


8.84 


6282 


47 




14 


0708 


8.30 


6459 




4249 


8.83 


575 1 


46 




15 


1206 


8.28 


6427 




4778 


8.82 


5222 


45 




16 


1703 


8.27 


6 395 


•53 


53o8 


8.81 


4692 


44 




17 


2199 


8.26 


6363 


•54 


5836 


8.80 


4164 


43 




18 


2695 


8.25 


6331 




6364 


8.79 


3636 


42 




19 


3*9* 


8.24 


6299 




6892 


8.78 


3108 


41 




20 
21 


3685 


8.23 


6266 




7419 


8.77 


2581 


40 
~39~ 




9-394*79 


8.22 


9.986234 




9.407945 


8.76 


10.592055 




22 


4 6 73 


8.21 


6202 




8471 


8-75 


1529 


38 




23 


5166 


8.20 


6169 




8997 


8-74 


1003 


37 




24 


5658 


8.20 


6137 




9.409521 


8.74 


10.590479 


36 




25 


6150 


8.18 


6104 




9.410045 


8.73 


10.589955 


35 




26 


6641 


8.17 


6072 




0569 


8.72 


943 " 


34 




27 


7132 


8.17 


6039 




1092 


8.71 


8908 


33 




28 


7621 


8.16 


6007 




1615 


8.70 


8385 


32 




29 


8111 


8.15 


5974 




2137 


8.69 


7863 


31 




30 
31 


8600 


8.14 


5942 


•54 
•55 


2658 


8.68 


734 2 


30 
29 




9088 


8.13 


9.985909 


9-4i 3 1 79 


8.67 


10.586821 




32 


9-399575 


8.12 


5876 




3699 


8.66 


6301 


28 




33 


9.400062 


8.11 


5 8 43 




4219 


8.65 


5781 


27 




34 


0549 


8.10 


5811 




4738 


8.64 


5262 


26 




35 


i°35 


8.09 


5778 




5 2 57 


8.64 


4743 


25 




36 


1520 


8.08 


5745 




5775 


8.63 


4225 


24 




37 


2005 


8.07 


57i2 




6293 


8.62 


3707 


23 




38 


2489 


8.06 


5679 




6810 


8.61 


3190 


22 




39 


2972 


8.05 


5646 




7326 


8.60 


2674 


21 




40 
41 


3455 


8.04 


5613 




7842 


8.59 


2158 


20 
19 




9.403938 


8.03 


9.985580 


9.418358 


8.58 


10.581642 




42 


4420 


8.02 


5547 




8873 


8.57 


1127 


18 




43 


4901 


8.01 


55H 




9387 


8.56 


0613 


17 




44 


53^ 


8.00 


5480 




9.419901 


8-55 


10.580099 


16 




45 


5862 


7-99 


5447 


•55 


9.420415 


8.55 


10.579585 


15 




46 


6341 


7.98 


54H 


.56 


0927 


8.54 


9073 


14 




47 


6820 


7-97 


5380 




1440 


8.53 


8560 


13 




48 


7299 


7.96 


5347 




1952 


8.52 


8048 


12 




49 


7777 


7-95 


53*4 




2463 


8.51 


7537 


11 




50 
51 


8254 


7-94 


5280 




2974 


8.50 
8.49 


7026 


10 

9 




9.408731 


7-94 


9.985247 




9.423484 


10.576516 




b'Z 


9207 


7-93 


5213 




3993 


8.48 


6007 


8 




53 


9.409682 


7.92 


5180 




45°3 


8.48 


5497 


7 




54 


9.410157 


7.91 


5146 




5011 


8.47 


4989 


6 




55 


0632 


7.90 


5113 




5519 


8.46 


4481 


5 




56 


1106 


7.89 


5°79 




6027 


8.45 


3973 


4 




57 


1579 


7.88 


5°45 




6534 


8.44 


3466 


3 




58 


2052 


7.87 


5011 




7041 


8.43 


2959 


2 




59 


2524 


7.86 


4978 


.56 


7547 


8-43 


^453 


1 




60 


9.412996 




9.984944 


Diff.l" 


9.428052 
Cotang. 




10.571948 



M. 




Cosine. 


Diff. 1" 


Sine. 


Diff. 1" 


Tang. 




104° 








75° 





56 



15° 


SINES AND TAMTaENTTS. 


-' 

164° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 
IO.571948 


60 ! 


9.412996 


7.85 


9.984944 


-SI 


9.428052 


8.42 


1 


34 6 7 


7.84 


4910 




8557 


8.41 


1443 


59 i 


2 


3938 


7.83 


4876 




9062 


8.40 


0938 


58 1 


3 


4408 


7.83 


4842 




9.429566 


8-39 


IO.570434 


57 ! 


4 


4878 


7.82 


4808 




9.430070 


8.38 


IO.569930 


56 1 


5 


5347 


7.81 


4774 




0573 


8.38 


9427 


55 1 


6 


5815 


7.80 


4740 




1075 


8.37 


8925 


54 ; j 


7 


6283 


7-79 


4706 




1577 


8.36 


8423 


53 |i 


8 


6751 


7.78 


4672 




2079 


8.35 


7921 


52 ! 


y 


7217 


7-77 


4637 




2580 


8-34 


7420 


M | 


10 

n 


7684 
9.418150 


7.76 

7-75 


4603 




3080 


8-33 


6920 
IO.566420 


50 ! 
49 


9.984569 




9.433580 


8.32 


12 


8615 


7-74 


4535 




4080 


8.32 


5920 


48 


13 


9079 


7-73 


4500 




4579 


8.31 


54^1 


47 


14 


9.419544 


7-73 


4466 


•SI 


5078 


8.30 


4922 


46 


15 


9.420007 


7.72 


443 2 


•58 


5576 


8.29 


4424 


45 


16 


0470 


7.71 


4397 




6073 


8.28 


39 2 7 


44 


IV 


0933 


7.70 


4363 




6570 


8.28 


343° 


43 


18 


1395 


7.69 


4328 




7067 


8.27 


2 933 


42 


19 


1857 


7.68 


4294 




75 6 3 


8.26 


2 437 


41 


20 
21 


2318 


7.67 


4259 




8059 


8.25 


1941 
10.561446 


40 
39 


9.422778 


7.67 


9.984224 




9.438554 


8.24 


22 


3238 


7.66 


4190 




9048 


8.23 


0952 


38 


23 


3 6 97 


7.65 


4155 




9-439543 


8.23 


10.560457 


37 


24 


4156 


7-64 


4120 




9.440036 


8.22 


10.559964 


36 


25 


4615 


7-63 


4085 




0529 


8.21 


947i 


35 


26 


5°73 


7.62 


4050 




1022 


S,20 


8978 


34 


27 


553° 


7.61 


4015 




1514 


*>.*9* 


8486 


33 


28 


5987 


7.60 


398i 




2006 


&.19 


7994 


32 


29 


6 443 


7.60 


3946 




2497 


8.18 


7503 


31 


30 
31 


6899 
9-4 2 7354 


7-59 


39 11 




2988 


8.17 


7012 


30 

29 


7.58 


9.983875 


•58 


9-443479 


8.16 


10.556521 


32 


7809 


7-57 


3840 


•59 


3968 


8.16 


6032 


28 


33 


8263 


7.56 


3805 




4458 


8.15 


554* 


27 


34 


8717 


7-55 


377° 




4947 


8.14 


5°53 


26 


35 


9170 


7-54 


3735 




5435 


8.13 


45 6 5 


25 


36 


9.429623 


7-54 


3700 




59 2 3 


8.12 


4077 


24 


37 


9.430075 


7-53 


3664 




641 1 


8.12 


3589 


23 


38 


0527 


7.52 


3629 




6898 


8.11 


3102 


22 


39 


0978 


7-5 1 


3594 




7384 


8.10 


2616 


21 


40 
41 


1429 


7.50 


3558 




7870 


8.09 


2130 

10.551644 


20 
19 


9.431879 


7-49 


9.983523 


9.448356 


8.09 


42 


2329 


7-49 


3487 




8841 


8.08 


1159 


18 


43 


2778 


7.48 


3452 




9326 


8.07 


0674 


17 


44 


3226 


7-47 


3416 




9.449810 


8.06 


10.550190 


16 


4o 


3 6 75 


7.46 


338i 




9.450294 


8.06 


10.549706 


15 


46 


4122 


7-45 


3345 




0777 


8.05 


9223 


14 


47 


4569 


7-44 


33°9 


•59 


1260 


8.04 


8740 


13 


48 


5016 


7-44 


3273 


.60 


1743 


8.03 


8257 


12 


49 


5462 


7-43 


3238 




2225 


8.C2 


7775 


11 


50 
51 


5908 


7.42 


3202 




2706 


8.02 


7294 
10.546813 


10 

9 


9-43 6 353 


7.41 


9.983166 




9-453 l8 7 


8.01 


52 


6798 


7.40 


3*3° 




3668 


8.00 


633 2 


8 


53 


7242 


7.40 


3°94 




4148 


7-99 


585 2 


t 


54 


7686 


7-39 


3058 




4628 


7-99 


5372 


6 


65 


8129 


7-38 


3022 




5 io 7 


7.98 


4893 


1 


56 


8572 


7-37 


2986 




5586 


7-97 


4414 


4 


67 


9014 


7-3 6 


2950 




6064 


7.96 


393 6 


3 


58 


9456 


7.36 


2914 




6542 


7.96 


3458 


2 


59 


9-439 8 97 


7-35 


2878 


.60 


7019 


7-95 


2981 


1 


60 


9.440338 




9.982842 




9.457496 


Diff. 1" 


10.542504 
Tang. 



M. 


Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


105° 








74° 



57 



16° 




LOGARITHMIC 




163° | 




J M. 



Sine. 


Diff. 1" 


Cosine. 


Diff. 1" 


Tang. 


Diff. 1" 


Cotang. 


60 




9.440338 


7-34 


9.982842 


.60 


9.457496 


7-94 


10.542504 




1 


0778 


7-33 


2805 


.60 


7973 


7-93 


2027 


59 




2 


1218 


7-3 2 


2769 


.61 


8449 


7-93 


1551 


58 




3 


1658 


7.31 


2733 




8925 


7.92 


1075 


57 




4 


2096 


7-3 1 


2696 




9400 


7.91 


0600 


56 




5 


2535 


7-3° 


2660 




9-459 8 75 


7.90 


IO.540125 


55 




6 


2973 


7.29 


2624 




9.460349 


7.90 


IO.539651 


54 




7 


3410 


7.28 


2587 




0823 


7.89 


9177 


53 




8 


3 8 47 


7.27 


2551 




1297 


7.88 


8703 


52 




9 


4284 


7.27 


2514 




1770 


7.88 


8230 


51 




10 
11 


4720 


7.26 


2477 




2242 


7.87 


7758 


50 
49 




9-445I55 


7.25 


9.982441 




9.462714 


7.86 


10.537286 




12 


559° 


7.24 


2404 




3186 


7.85 


6814 


48 




13 


6025 


7.23 


2367 




3658 


7.85 


6342 


47 




14 


6459 


7-^3 


2331 




4129 


7.84 


5871 


46 




15 


6893 


7.22 


2294 




4599 


7-83 


5401 


45 




16 


7326 


7.21 


2257 


.61 


5069 


7.83 


4931 


44 




17 


7759 


7.20* 


2220 


.62 


5539 


7.82 


4461 


43 




18 


8191 


7.20 


2183 




6008 


7.81 


3992 


42 




19 


8623 


7.19 


2146 




6476 


7.80 


35 2 4 


41 




20 
21 


9054 
9485 


7.18 
7.17 


2109 




_6?45 

9.467413 


7.80 


3°55 


40 
39 




9.982072 




7-79 


10.532587 




22 


9-4499 x 5 


7.16 


2035 




7880 


7.78 


2120 


38 




23 


9-45°345 


7.16 


1998 




8347 


7.78 


1653 


37 




24 


°775 


7-15 


1961 




8814 


7-77 


1186 


36 




25 


1204 


7.14 


1924 




9280 


7.76 


0720 


35 




26 


1632 


7-13 


1886 




9.469746 


7-75 


10.530254 


34 




27 


2060 


7-13 


1849 




9.47021 1 


7-75 


10.529789 


33 




28 


2488 


7.12 


1812 




0676 


7-74 


2P 4 


32 




29 


2915 


7.11 


1774 




1 141 


7-73 


8859 


31 1 




30 
31 


3342 
9.453768 


7.10 


1737 


.62 


1605 


7-73 


8395 


30 
29 




7.10 


9.981699 


.63 


9.472068 


7.72 


10.527932 




32 


4194 


7.09 


1662 




*53 2 


7.71 


7468 


28 




33 


4619 


7.08 


1625 




2995 


7.71 


7005 


27 




34 


5044 


7.07 


1587 




3457 


7.70 


6543 


26 




35 


5469 


7.07 


1549 




39 J 9 


7.69 


6081 


25 




36 


5 8 93 


7.06 


1512 




438i 


7.69 


5619 


24 




37 


6316 


7.05 


1474 




4842 


7.68 


5158 


23 




38 


6739 


7.04 


1436 




53°3 


7.67 


4697 


22 




39 


7162 


7.04 


1399 




5763 


7.67 


4237 


21 




40 
41 


7584 
9.458006 


7-°3 


1361 




6223 


7.66 


3777 


20 
19 




7.02 


9.981323 




9.476683 


7.65 


10.523317 




42 


8427 


7.01 


1285 




7142 


7.65 


2858 


18 




43 


8848 


7.01 


1247 




7601 


7.64 


2399 


17 




44 


9268 


7.00 


1209 




8059 


7.63 


1941 


16 




45 


9.459688 


6.99 


1171 


.63 


8517 


7- 6 3 


1483 


15 




46 


9.460108 


6.98 


"33 


.64 


8975 


7.62 


1025 


14 




47 


0527 


6.98 


1095 




9432 


7.61 


0568 


13 




48 


0946 


6.97 


1057 




9.479889 


7.61 


10.520m 


12 




49 


1364 


6.96 


1019 




9-4 8o 345 


7.60 


10.519655 


11 




50 
~51 


1782 


6.95 
6.95 


0981 




0801 


7-59 


9199 


10 
9 




9.462199 


9.980942 




9.481257 


7-59 


10.518743 




52 


2616 


6.94 


0904 




1712 


7.58 


8288 


8 




53 


3°3 2 


6-93 


0866 




2167 


7-57 


7833 


7 




54 


3448 


6.93 


0827 




2621 


7-57 


7379 


6 




55 


3864 


6.92 


0789 




3°75 


7.56 


6925 


5 




56 


4279 


6.91 


0750 




3529 


7-55 


6471 


4 




57 


4694 


6.90 


0712 




3982 


7-55 


6018 


3 




58 


5108 


6.9c 


0673 




4435 


7-54 


5565 


2 




59 


55 22 


6.89 


0635 


.64 


4887 


7-53 


5113 


1 




60 


9465935 
Cosine. 




9.980596 


Diff.]' 


9.485339 




10.514661 



M. 




Diff. 1" 


Sine. 


Cotang. 


Diff. 1" 


Tang. 




106° 








73° 





58 



17° 


SINES AND TANCHSNTS. 


162° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Oiff.l" 


Tang. 


Diff. 1" 
7-53 


Cotang. 


60 


9.465935 


6.88 


9.980596 


.64 


9-4 8 5339 


IO.514661 


1 


6348 


6-88 


0558 


.64 


579i 


7.52 


4209 


59 


2 


6761 


6.87 


0519 


.65 


6242 


7-5i 


375 8 


58 


3 


7173 


6.86 


0480 




6693 


7-5i 


33°7 


57 


4 


75^5 


6.85 


0442 




7H3 


7.50 


2857 


56 


5 


7996 


6.85 


0403 




7593 


7-49 


2407 


55 


6 


8407 


6.84 


0364 




8043 


7-49 


1957 


54 


7 


8817 


6.83 


0325 




8492 


7.48 


1508 


53 


8 


9227 


6.83 


0286 




8941 


7-47 


1059 


52 


9 


9.469637 


6.82 


0247 




9390 


7-47 


0610 


51 


10 
11 


9.470046 


6.81 


0208 




9.489838 


7.46 


10.510162 


50 
49 


0455 


6.80 


9.980169 




9.490286 


7.46 


10.509714 


12 


0863 


6.80 


0130 




o733 


7-45 


9267 


48 


13 


1271 


6.79 


0091 




1180 


7-44 


8820 


47 


14 


1679 


6.78 


0052 




1627 


7-44 


8373 


46 


15 


2086 


6.78 


9.980012 




2073 


7-43 


7927 


45 


16 


2492 


6.77 


9-979973 


.65 


2519 


7-43 


7481 


44 


17 


2898 


6.76 


9934 


.66 


2965 


7.42 


7035 


43 


18 


33°4 


6.76 


9895 




3410 


7.41 


6590 


42 


19 


3710 


6.75 


9855 




3 8 54 


7.40 


6146 


41 


2U 
21 


4115 


6.74 


9816 




4299 


7.40 


5701 


40 
39 


9.474519 


6.74 


9.979776 




9-494743 


7-39 


10.505257 


22 


49 2 3 


6.73 


9737 




5186 


7-39 


4814 


38 


23 


53 2 7 


6.72 


9697 




5630 


7-38 


437° 


37 


24 


5730 


6.72 


9658 




6073 


7-37 


39 2 7 


36 


25 


6l 33 


6.71 


9618 




65!5 


7-37 


34 8 5 


35 


26 


6536 


6.70 


9579 




6957 


7-3 6 


3°43 


34 


27 


6938 


6.69 


9539 




7399 


7.36 


2601 


33 


28 


734° 


6.69 


9499 




7841 


7-35 


2159 


32 


29 


774i 


6.68 


9459 




8282 


7-34 


1718 


31 


30 
31 


8142 


6.67 


9420 




8722 


7-34 


1278 
0837 


30 

29 


9.478542 


6.67 


9.979380 




9163 


7-33 


32 


8942 


6.66 


934° 


.66 


9.499603 


7-33 


10.500397 


28 


33 


9342 


6.65 


9300 


.67 


9.500042 


7.32 


10.499958 


27 


34 


9.479741 


6.65 


9260 




0481 


7-3 1 


95i9 


26 


35 


9.480140 


6.64 


9220 




0920 


7.31 


9080 


25 


36 


°539 


6.63 


9180 




1359 


7.30 


8641 


24 


37 


0937 


6.63 


9140 




1797 


7-3° 


8203 


23 


38 


1334 


6.62 


9100 




2235 


7.29 


7765 


22 


39 


1731 


6.61 


9059 




2672 


7.28 


7328 


21 


40 
41 


2128 


6.61 


9019 




3109 


7.28 


6891 


20 
19 


9.482525 


6.60 


9.978979 




9.503546 


7.27 


10.496454 


42 


2921 


6.59 


8939 




3982 


7.27 


6018 


18 


43 


3316 


6.59 


8898 




4418 


7.26 


55^ 


17 


44 


3712 


6.58 


8858 




4854 


7.25 


5H6 


16 


45 


4107 


6.57 


8817 




5289 


7.25 


471 1 


15 


46 


4501 


6.57 


8777 




5724 


7.24 


4276 


14 


47 


4895 


6.56 


8736 


.67 


6159 


7.24 


3841 


13 


48 


5289 


6.55 


8696 


.68 


6593 


7.23 


34°7 


12 


49 


5682 


6.55 


8655 




7027 


7.22 


2973 


11 


50 
51 


6075 
9.486467 


6-54 


8615 




7460 
9.507893 


7.22 


2540 


10 
9 


6-53 


9.978574 


7.21 


10.492107 


52 


6860 


6-53 


8533 




8326 


7.21 


1674 


8 


53 


7251 


6.52 


8493 




8759 


7.20 


1 241 


7 


54 


7643 


6.51 


8452 




9191 


7.19 


0809 


6 


55 


8034 


6.51 


841 1 




9.509622 


7.19 


10.490378 


5 


56 


8424 


6.50 


8370 




9.510054 


7.18 


10.489946 


4 


57 


8814 


6.50 


8329 




0485 


7.18 


9515 


3 


58 


9204 


6.49 


8288 




0916 


7.17 


9084 


2 


59 


9593 


6.48 


8247 


.68 


1346 


7.17 


8654 


1 


60 


9.489982 




9.978206 
Sine. 


Diff.l' 


9.511776 




10.488224 



M. 


Cosine. 


Diff. 1" 


Cotang. 


Diff. 1" 


Tang. 


107° 








72° 



59 



18° 




LOGARITHMIC 




161° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Dift.l" 
.68 


Tang. 


Diff. 1" 


Cotang. 


60 


9.489982 


6.48 


9.978206 


9.511776 


7.16 


IO.488224 


1 


9.490371 


6.47 


8165 




2206 


7.16 


7794 


59 


2 


0759 


6.46 


8124 


.68 


2635 


7-15 


7365 


58 


3 


1 147 


6.46 


8083 


.69 


3064 


7.14 


6936 


57 


4 


1535 


6.45 


8042 




3493 


7.14 


6507 


56 


5 


1922 


6.44 


8001 




3921 


7.13 


6079 


55 


6 


2308 


6.44 


7959 




4349 


7.13 


5 6 5i 


54 


7 


2695 


6.43 


7918 




4777 


7.12 


5223 


53 


8 


3081 


6.42 


7877 




5204 


7.12 


4796 


52 


9 


3466 


6.42 


7835 




5 6 3! 


7.11 


43 6 9 


51 


10 
11 


3851 


6.41 


7794 
9.977752 




6057 


7.10 


3943 


50 
49 


9.494236 


6.41 




9.516484 


7.10 


10.483516 


12 


4621 


6.40 


7711 




6910 


7.09 


3090 


48 


13 


5°°5 


6.39 


7669 




7335 


7.09 


2665 


47 


14 


5388 


6.39 


7628 




7761 


7.08 


2239 


46 


15 


5772 


6.38 


7586 


.69 


8185 


7.08 


1815 


45 


16 


6154 


6.37 


7544 


.70 


8610 


7.07 


1390 


44 


17 


6537 


, 6 -37 


.7503 




9034 


7.06 


0966 


43 


18 


6919 


6.36 


7461 




9458 


7.06 


0542 


42 


19 


7301 


6.36 


7419 




9.519882 


7.05 


10.480118 


41 


20 
21 


7682 


6.35 


7377 




9.520305 


7-05 


10.479695 


40 
39 


9.498064 


6-34 


9-977335 


0728 


7.04 


9272 


22 


8444 


6.34 


7293 




"5i 


7.04 


8849 


38 


23 


8825 


6-33 


7251 




1573 


7-03 


8427 


37 


24 


9204 


6.32 


7209 




1995 


7.03 


8005 


36 


25 


9584 


6.32 


7167 




2417 


7.02 


7583 


35 


26 


9.499963 


6.31 


7125 




2838 


7.02 


7162 


34 


27 


9.500342 


6.31 


7083 




3 2 59 


7.01 


6741 


33 


28 


0721 


6.30 


7041 




3680 


7.OI 


6320 


32 


29 


1099 


6.29 


6999 




4100 


7.00 


5900 


31 


30 
31 


1476 


6.29 


6957 




4520 


6.99 


5480 


30 
29 


9.501854 


6.28 


9.976914 


.70 


9.524939 


6.99 


10.475061 


32 


2231 


6.28 


6872 


•71 


5359 


6.98 


4641 


28 


33 


2607 


6.27 


6830 




5778 


6.98 


4222 


27 


34 


2984 


6.26 


6787 




6197 


6.97 


3803 


26 


35 


3360 


6.26 


6745 




6615 


6.97 


3385 


25 


36 


3735 


6.25 


6702 




7°33 


6.96 


2967 


24 


37 


4110 


6.25 


6660 




745 x 


6.96 


2549 


23 


38 


4485 


6.24 


6617 




7868 


6.95 


2132 


22 


39 


4860 


6.23 


6574 




8285 


6.95 


1715 


21 


40 
41 


5*34 


6.23 


6532 




8702 


6.94 


1298 


20 
19 


9.505608 


6.22 


9.976489 


9.529119 


6-93 


10.470881 


42 


598i 


6.22 


6446 




9535 


6-93 


0465 


18 


43 


6 354 


6.21 


6404 




9.529950 


6 -93 


10.470050 


17 


44 


6727 


6.20 


6361 




9.530366 


6.92 


10.469634 


16 


45 


7099 


6.20 


6318 




0781 


6.91 


9219 


15 


46 


7471 


6.19 


6275 


•71 


1196 


6.91 


8804 


14 


47 


7843 


6.19 


6232 


.72 


1611 


6.90 


8389 


13 


48 


8214 


6.18 


6189 




2025 


6.90 


7975 


12 


49 


8585 


6.18 


6146 




2439 


6.89 


7561 


11 


50 
51 


8956 
9326 


6.17 


6103 




2853 


6.89 


7H7 


10 
9 


6.16 


9.976060 


9.533266 


6.88 


10.466734 


52 


9.509696 


6.16 


6017 




3 6 79 


6.88 


6321 


8 


53 


9.510065 


6.15 


5974 




4092 


6.87 


5908 


7 


54 


0434 


6.15 


593° 




4504 


6.87 


5496 


6 


55 


0803 


6.14 


.5887 




4916 


6.86 


5084 


5 


56 


1172 


6.13 


5844 




53*8 


6.86 


4672 


4 


57 


1540 


6.13 


5800 




5739 


6.85 


4261 


3 


58 


1907 


6.12 


5757 




6150 


6.85 


3850 


2 


59 


2275 


6.12 


57H 


.72 


6561 


6.84 


3439 


1 


60 


9.512642 
Cosine. 




9.975670 


Diff.l" 


9.536972 
Cotang. 




10.463028 




M. 


Diff. 1" 


I Sine. 


Diff. 1" 


Tang. 


108° 










71° 



GO 



19° 


SINES AND TANCtENTS. 


160° 




M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.r' 


Tang. 


Diff. 1" 


Cotang. 


60 




9.512642 


6.11 


9.975670 


•73 


9.536972 


6.84 


10.463028 




1 


3009 


6.II 


5627 




7382 


6.83 


2618 


59 




2 


3375 


6.10 


5583 




7792 


6.83 


2208 


58 




3 


374i 


6.09 


5539 




8202 


6.82 


1798 


57 




4 


4107 


6.09 


5496 




86ll 


6.82 


1389 


56 




5 


4472 


6.08 


545* 




9020 


6.81 


0980 


65 




6 


4837 


6.08 


5408 




9429 


6.81 


0571 


54 




7 


5202 


6.07 


53 6 5 




9-539837 


6.80 


IO.460163 


63 




8 


5566 


6.07 


53 21 




9.540245 


6.80 


IO.459755 


52 




y 


593° 


6.06 


5277 




0653 


6.79 


9347 


51 




10 

n 


6294 


6.05 


5233 




I061 


6.79 


8939 


50 
49 




9.516657 


6.05 


9.975189 




9.541468 


6.78 


10.458532 




12 


7020 


6.04 


5H5 




1875 


6.78 


8125 


48 




13 


7382 


6.04 


5101 




2281 


6.77 


7719 


47 




14 


7745 


6.03 


5°57 




2688 


6.77 


7312 


46 




15 


8107 


6.03 


5 OI 3 


•73 


3°94 


6.76 


6906 


45 




16 


8468 


6.02 


4969 


•74 


3499 


6.76 


6501 


44 




17 


8829 


6.01 


4925 




3905 


6.75 


6095 


43 




18 


9190 


6.01 


4880 




4310 


6.75 


5690 


42 




19 


955i 


6.00 


4836 




4715 


6.74 


5285 


41 




20 
21 


9.519911 


6.00 


4792 




5"9 


6.74 


' 4881 


40 
39 




9.520271 


5-99 


9.974748 




9-545524 


6-73 


10.454476 




22 


0631 


5-99 


4703 




5928 


6.73 


4072 


38 




23 


0990 


5-98 


4659 




6331 


6.72 


3669 


37 




24 


1349 


5.98 


4614 




6735 


6.72 


3265 


36 




25 


1707 


5-97 


457o 




7138 


6.71 


2862 


35 




26 


2066 


5-96 


4525 




754o 


6.71 


2460 


34 




27 


2424 


5-9 6 


4481 




7943 


6.70 


2057 


33 




28 


2781 


5-95 


443 6 




8345 


6.70 


1655 


32 




29 


3138 


5-95 


4391 


•74 


8747 


6.69 


1253 


31 




30 
31 


3495 


5-94 


4347 


•75 


9149 


6.69 


0851 


30 

29 




9.523852 


5-94 


9.974302 




955° 


6.68 


0450 




32 


4208 


5-93 


4 2 57 




9.549951 


6.68 


10.450049 


28 




33 


4564 


5-93 


4212 




9-55°35 2 


6.67 


10.449648 


27 




34 


4920 


5.92 


4167 




0752 


6.67 


9248 


26 




35 


52-75 


5.91 


4122 




1152 


6.66 


8848 


25 




36 


5630 


5.91 


4077 




1552 


6.66 


8448 


24 




37 


59 8 4 


5-9° 


4032 




1952 


6.65 


8048 


23 




38 


6 339 


5-9° 


3987 




2351 


6.65 


7649 


22 




39 


6693 


5-89 


3942 




2750 


6.65 


7250 


21 




40 
41 


7046 


5.89 


3897 




3*49 


6.64 


6851 


20 
19 




9.527400 


5.88 


9.973852 


9-553548 


6.64 


10.446452 




42 


7753 


5.88 


3807 




394 6 


6.63 


6054 


18 




43 


8105 


5.87 


3761 


•75 


4344 


6.63 


5656 


17 




44 


8458 


5.87 


3716 


.76 


474 1 


6.62 


5 2 59 


16 




45 


8810 


5.86 


3671 




5 J 39 


6.62 


4861 


15 




46 


9161 


5.86 


3625 




5536 


6.61 


4464 


14 




47 


9513 


5.85 


3580 




5933 


6.61 


4067 


13 




48 


9.529864 


5.85 


3535 




6329 


6.60 


3671 


12 




49 


9.530215 


5.84 


3489 




6725 


6.60 


32-75 


11 




60 
51 


o5 6 5 


5.84 


3444 




7121 


6-59 


2.879 


10 

9 




9-53°9 I 5 


5-83 


9-973398 


9-5575I7 


6.59 


10.442483 




52 


1265 


5.82 


335 2 




7913 


6.59 


2087 


8 




53 


1614 


5.82 


33°7 




8308 


6.58 


1692 


7 




54 


1963 


5.81 


3261 




8702 


6.58 


1298 


6 




55 


2312 


5-8i 


3 2I 5 




9097 


6.57 


0903 


6 




56 


2661 


5.80 


3169 




9491 


6.57 


0509 


4 




57 


3009 


5.80 


3 I2 4 




9.559885 


6.56 


10.4401 1 5 


3 




58 


3357 


5-79 


3078 


.76 


9.560279 


6.56 


10.439721 


2 




59 


3704 


5-79 


3032 


•77 


0673 


6.55 


9327 


1 




60 


9.534052 
Cosine. 




9.972986 


Diff.l" 


9.561066 
Cotang. 




10.438934 



M. 




Diff. 1" 


Sine. 


Diff. 1" 


Tang. 




109° 








70° 





61 



20° 




LOGARITHMIC 




159° 




M. 
~0 


Sine. 


Diff. 1" 


Cosine. 


Diff.1" 


Tang. 


Diff 1" 


Cotang. 


1 

60 




9.534052 


5-78 


9.972986 


•77 


9.561066 


6.55 


IO -43 8 934 




1 


4399 


5-78 


2940 




1459 


6.54 


8541 


o9 




2 


4745 


5-77 


2894 




1851 


6.54 


8149 


58 




3 


5092 


5-77 


2848 




2244 


6-53 


7756 


57 




4 


5438 


5-76 


2802 




2636 


6-53 


73 6 4 


56 




5 


5783 


5.76 


2755 




3028 


6-53 


6972 


55 




6 


6129 


5-75 


2709 




3419 


6.52 


6581 


54 




V 


6474 


5-74 


2663 




3811 


6.52 


6189 


53 




8 


6818 


5-74 


2617 




4202 


6.51 


5798 


52 




y 


7163 


5-73 


2570 




4592 


6.51 


5408 


51 




10 

n 


7507 


5-73 


2524 




4983 
9-5 6 5373 


6.50 


5 OI 7 


50 
49 




9-53785I 


5.72 


9.972478 


•77 


6.50 


10.434627 




12 


8194 


5-7 2 


2431 


.78 


57 6 3 


6.49 


4 2 37 


48 




13 


8538 


5-7i 


2385 




6i53 


6.49 


3847 


47 




14 


8880 


5-7i 


2338 




6542 


6.49 


3458 


46 




15 


9223 


5-7o 


2291 




6932 


6.48 


3068 


45 




16 


9565 


5-7° 


2245 




7320 


6.48 


2680 


44 




IV 


9-5399°7 


.5- 6 9 


2198 




7709 


6.47 


2291 


43 




18 


9.540249 


5- 6 9 


2151 




8098 


6.47 


1902 


42 




19 


0590 


5.68 


2105 




8486 


6.46 


1514 


41 




20 
~21 


0931 


5.68 


2058 




8873 


6.46 


1127 


40 
39 




9.541272 


5.67 


9.972011 




9261 


6.45 


0739 




22 


1613 


5- 6 7 


1964 




9.569648 


6.45 


10.430352 


38 




23 


1953 


5.66 


1917 




9.570035 


6.45 


10.429965 


3Y 




24 


2293 


5.66 


1870 




0422 


6.44 


9578 


36 




25 


2632 


5.65 


1823 




0809 


6.44 


9191 


35 




26 


2971 


5.65 


1776 


.78 


1195 


6.43 


8805 


34 




27 


3310 


5- 6 4 


I729 


•79 


1581 


6 -43 


8419 


33 




28 


3 6 49 


5- 6 4 


1682 




1967 


6.42 


8033 


32 




29 


39 8 7 


5-63 


1635 




2352 


6.42 


7648 


31 




30 
31 


4325 


5-63 


1588 




2738 
9-573 I2 3 


6.42 
6.41 


7262 


30 
29 




9.544663 


5.62 


9.971540 




10.426877 




32 


5000 


5.62 


1493 




3507 


6.41 


6 493 


28 




33 


533 8 


5.61 


1446 




3892 


6.40 


6108 


2V 




34 


5 6 74 


5.61 


1398 




4276 


6.40 


57 2 4 


26 




3b 


6011 


5.60 


1351 




4660 


6 -39 


534° 


25 




36 


6347 


5.60 


I303 




5044 


6-39 


4956 


24 




37 


6683 


5-59 


1256 




5427 


6.39 


4573 


23 




38 


7019 


5-59 


1208 




5810 


6.38 


4190 


22 




39 


7354 


5-58 


Il6l 




6193 


6.38 


3807 


21 




40 
41 


7689 
9.548024 


5.58 


III3 


•79 
.80 


6576 


6-37 


3424 
10.423042 


20 
19 




5-57 


9.971066 


9.576958 


6.37 




42 


8359 


5-57 


1018 




734i 


6.36 


2659 


18 




43 


8693 


5.56 


0970 




77 2 3 


6.36 


2277 


IV 




44 


9027 


5.56 


0922 




8104 


6.36 


1896 


16 




45 


9360 


5-55 


0874 




8486 


6-35 


1514 


15 




46 


9-549 6 93 


5-55 


0827 




8867 


6-35 


"33 


14 




4V 


9.550026 


5-54 


0779 




9248 


6-34 


0752 


13 




48 


0359 


5-54 


0731 




9.579629 


6-34 


10.420371 


12 




49 


0692 


5-53 


0683 




9.580009 


6-34 


10.419991 


11 




50 
51 


1024 


5-53 
5-5* 


0635 




0389 


6 -33 


961 1 

10.419231 


10 
9 




9-55 J 35 6 


9.970586 




9.580769 


6 -33 




52 


1687 


5-52 


0538 




1 149 


6.32 


8851 


8 




53 


2018 


5.52 


0490 




1528 


6.32 


8472 


7 




54 


2349 


5-5i 


0442 




1907 


6.32 


8093 


6 




55 


2680 


5-5i 


. 0394 


.80 


2286 


6.31 


7714 


6 




56 


3010 


5-5° 


°345 


.81 


2665 


6.31 


7335 


4 




5V 


334i 


5.50 


0297 




3°43 


6.30 


6957 


3 




58 


3670 


5-49 


0249 




3422 


6.30 


6578 


2 




59 


4000 


5-49 


0200 


.81 


3800 


6.29 


6200 


1 




60 


9.554329 




9.970152 


Diff.l" 


9.584177 




10.415823 




i 




Cosine. 


Diff. 1" 


Sine. 


Cotang. 


Diff. 1" 


Tang. 




110° 








69° 





62 



21° 


SINES AND TANGENTS. 


158° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9-5543^9 


5.48 


9.970152 


.81 


9.584177 


6.29 


10.415823 


1 


4658 


5-48 


0103 




4555 


6.29 


5445 


59 


2 


4987 


5-47 


0055 




4932 


6.28 


5068 


58 


3 


53*5 


5-47 


9.970006 




53°9 


6.28 


4691 


57 


4 


5643 


5.46 


9.969957 




5686 


6.27 


43 J 4 


56 


5 


597i 


5-4 6 


9909 




6062 


6.27 


3938 


55 


6 


6299 


5-45 


9860 




6439 


6.27 


35 61 


54 


7 


6626 


5-45 


9811 




6815 


6.26 


3185 


53 


8 


6953 


5-44 


9762 




7190 


6.26 


2810 


52 


y 


7280 


5-44 


97H 




7566 


6.25 


2434 


51 


10 

n 


7606 


5-43 


9665 


.81 


7941 
9.588316 


6.25 


2059 


50 
49 


9-S5793 2 - 


5-43 


9.969616 


.82 


6.25 


10.411684 


12 


8258 


5-43 


9567 




8691 


6.24 


1309 


48 


13 


8583 


5-4* 


9518 




9066 


6.24 


0934 


47 


14 


8909 


5.42 


9469 




9440 


6.23 


0560 


46 


16 


9234 


5.41 


9420 




9.589814 


6.23 


10.410186 


45 


16 


9558 


5.41 


9370 




9.590188 


6.23 


10.409812 


44 


17 


9.559883 


5-4° 


9321 




0562 


6.22 


9438 


43 


18 


9.560207 


5.40 


9272 




°935 


6.22 


9065 


42 


19 


0531 


5-39 


9223 




1308 


6.22 


8692 


41 


20 
21 


0855 


5-39 


9173 




1681 


6.21 


' 8319 


40 

39 


9.561178 


5.38 


9.969124 




9.592054 


6.21 


10.407946 


22 


1501 


5-38 


9075 




2426 


6.20 


7574 


38 


23 


1824 


5-37 


9025 




2798 


6.20 


7202 


37 


24 


2146 


5-37 


8976 


.82 


3 J 7! 


6.20 


6829 


36 


25 


2468 


5-3 6 


8926 


.83 


3542 


6.I9 


6458 


35 


26 


2790 


5-36 


8877 




39M 


6.I9 


6086 


34 


27 


3112 


5-3 6 


8827 




4285 


6.18 


5715 


33 


28 


3433 


5-35 


8777 




4656 


6.18 


5344 


32 


29 


3755 


5-35 


8728 




5027 


6.l8 


4973 


31 


30 
31 


4°75 


5-34 
5-34 


8678 
9.968628 




5398 


6.I7 


4602 
10.404232 


30 
29 


9.564396 




9.595768 


6.I7 


32 


4716 


5-33 


8578 




6138 


6.16 


3862 


28 


33 


5036 


5-33 


8528 




6508 


6.l6 


349 2 


27 


34 


535 6 


5-3 2 


8479 




6878 


6.16 


3122 


26 


35 


5676 


5-32 


8429 




7247 


6.I5 


2 753 


25 


36 


5995 


5-3i 


8379 




7616 


6.I5 


2384 


24 


37 


6314 


5-3 1 


8329 




7985 


6.I5 


2015 


23 


38 


6632 


5-3i 


8278 


.83 


8354 


6.I4 


1646 


22 


39 


6951 


5-3° 


8228 


.84 


8722 


6.I4 


1278 


21 


40 
41 


7269 


5-3° 


8178 
9.968128 




9091 
9459 


6.I3 


0909 


20 
19 


9.567587 


5.29 




6.I3 


0541 


42 


7904 


5.29 


8078 




9.599827 


6.I3 


10.400173 


18 


43 


8222 


5.28 


8027 




9.600194 


6.12 


10.399806 


17 


44 


8539 


5.28 


7977 




0562 


6.12 


9438 


16 


45 


8856 


5.28 


7927 




0929 


6.11 


9071 


15 


46 


9172 


5.27 


7876 




1296 


6.11 


8704 


14 


47 


9488 


5.27 


7826 




1662 


6.11 


8338 


13 


48 


9.569804 


5.26 


7775 




2029 


6.10 


7971 


12 


49 


9.570120 


5.26 


7725 




2395 


6.10 


7605 


11 


50 
51 


0435 
9.570751 


5- 2 5 


7674 




2761 
9.603127 


6.10 
6.09 


7239 
,10.396873 


10 
9 


5- 2 5 


9.967624 


52 


1066 


5- 2 4 


7573 


.84 


3493 


6.09 


6507 


8 


53 


1380 


5- 2 4 


7522 


.85 


3858 


6.09 


6142 


7 


54 


1695 


5- 2 3 


747i 




4223 


6.08 


5777 


6 


55 


2009 


5- 2 3 


7421 




4588 


6.08 


5412 


5 


56 


2 3 2 3 


5- 2 3 


737o 




4953 


6.07 


5°47 


4 


57 


2636 


5.22 


7319 




53*7 


6.07 


4683 


3 


58 


2950 


5.22 


7268 




5682 


6.07 


4318 


2 


59 


3263 


5.21 


7217 


.85 


6046 


6.06 


3954 


1 


60 


9-573575 




9.967166 


Diff.l" 


9.606410 
Cotang. 




10.393590 
Tang. 



M. 


Cosine. 


Diff. 1" 


Sine. 


Diff. 1" 


111° 








68° 



63 



22° 




LOGARITHMIC 




157° | 


|M.| 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


i 
60 ! 


9-573575 


5.21 


9.967166 


.85 


9.606410 


6.06 


IO.393590 


1 


3888 


5.20 


7115 




6773 


6.06 


3227 


59 


2 


4200 


5.20 


7064 




7137 


6.05 


2863 


58 


3 


4512 


5.19 


7013 




7500 


6.05 


2500 


57 


4 


4824 


5- J 9 


6961 




7863 


6.04 


2137 


56 


5 


5136 


5-J9 


6910 




8225 


6.04 


1775 


55 


6 


5447 


5.18 


6859 




8588 


6.04 


1412 


54 


7 


5758 


5.18 


6808 


.85 


8950 


6.03 


1050 


53 


8 


6069 


5- J 7 


6756 


.86 


9312 


6.03 


0688 


52 


9 


6379 


5-i7 


6705 




9.609674 


6.03 


10.390326 


51 


10 
11 


6689 


5.16 


6653 




9.610036 


6.02 


10.389964 


50 
49 


9.576999 


5.16 


9.966602 




0397 


6.02 


10.389603 


12 


73°9 


5.16 


6550 




0759 


6.02 


9241 


48 


13 


7618 


5-J5 


6499 




1 1 20 


6.01 


8880 


47 


14 


7927 


5-i5 


6447 




1480 


6.01 


8520 


46 


15 


8236 


5-H 


6 395 




1841 


6.OI 


8159 


45 


16 


8545 


5-H 


6344 




220I 


6.00 


7799 


44 


17 


8853 


- 5-i3 


6292 




2561 


6.00 


7439 


43 


18 


9162 


5-!3 


6240 




2921 


6.00 


7079 


42 


19 


9470 


5-i3 


6188 




3281 


5-99 


6719 


41 


20 
21 


9-579777 


5.12 


6136 


.86 


3641 
9.614000 


5-99 


6359 


40 
39 


9.580085 


5.12 


9.966085 


.87 


5.98 


10.386000 


22 


0392 


5-" 


6033 




4359 


5-98 


5641 


38 


23 


0699 


5" 


598i 




4718 


5-98 


5282 


37 


24 


1005 


5.11 


5928 




5077 


5-97 


4923 


36 


25 


1312 


5.10 


5876 




5435 


5-97 


45 6 5 


35 


26 


1618 


5.10 


5824 




5793 


5-97 


4207 


34 


27 


1924 


5-°9 


5772 




6151 


5-9 6 


3849 


33 


28 


2229 


5-°9 


5720 




6509 


5-9 6 


349i 


32 


29 


2535 


5-o9 


5668 




6867 


5-9 6 


3 I 33 


31 


30 
31 


2840 


5.08 


5615 




7224 


5-95 


2776 


30 
29 


9-5 8 3 J 45 


5.08 


9.965563 




9.617582 


5-95 


10.382418 


32 


3449 


5-°7 


55" 




7939 


5-95 


2061 


28 


33 


3754 


5.07 


5458 




8295 


5-94 


1705 


27 


34 


4058 


5.06 


5406 


.87 


8652 


5-94 


1348 


26 


35 


4361 


5.06 


5353 


.88 


9008 


5-94 


0992 


25 


36 


4665 


5.06 


53 01 




93 6 4 


5-93 


0636 


24 


37 


4968 


5-o5 


5248 




9.619721 


5-93 


10.380279 


23 


38 


5272 


5-°5 


5i95 




9.620076 


5-93 


10.379924 


22 


39 


5574 


5-°4 


5H3 




0432 


5.92 


9568 


21 


40 
41 


5877 


5.04 


5090 




0787 


5.92 


9213 


20 
19 


9.586179 


5-°3 


9.965037 


9.621142 


5.92 


10.378858 


42 


6482 


5-°3 


4984 




1497 


5-9i 


8503 


18 


43 


6783 


5-°3 


493i 




1852 


5-9 1 


8148 


17 


44 


7085 


5.02 


4879 




2207 


5.90 


7793 


16 


45 


7386 


5.02 


4826 




2561 


5-9° 


7439 


15 


46 


7688 


5.01 


4773 




2915 


5-9° 


7085 


14 


47 


7989 


5.01 


4719 


.88 


3269 


5-89 


6731 


13 


48 


' 8289 


5.01 


4666 


■89 


3623 


5.89 


6377 


12 


49 


8590 


5.00 


4613 




397 6 


5-89 


6024 


11 


50 
51 


8890 


5.00 


4560 




4330 


5.88 


5670 


10 
9 


9.589190 


4.99 


9.964507 


9.624683 


5.88 


i°-3753 I 7 


52 


9489 


4-99 


4454 




5036 


5.88 


4964 


8 


53 


9.589789 


4.99 


4400 




5388 


5.87 


4612 


7 


54 


9.590088 


4.98 


4347 




574i 


5.87 


4259 


6 


55 


0387 


4.98 


4294 




6093 


5.87 


39°7 


5 


56 


0686 


4-97 


4240 




6445 


5.86 


3555 


4 


57 


0984 


4-97 


4187 




6797 


5.86 


3203 


3 


58 


1282 


4-97 


4133 




7149 


5.86 


2851 


2 


59 


1580 


4.96 


4080 


.89 


7501 


5-85 


2499 


1 


60 


9.591878 
Cosine. 




9.964026 
Sine. 




9.627852 




10.372148 



M. 


Diff. 1" 


Diff.l" 


1 Cotang. 


Diff. 1" 


Tang. 


112° 








67° 



64 



23° 


SINES AtfD TANGENTS. 


156° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.591878 


4.96 


9.964026 


.89 


9.627852 


5.85 


IO.372148 


1 


2176 


4-95 


3972 


.89 


8203 


5.85 


1797 


59 


2 


2473 


4-95 


3919 


.89 


8554 


5.85 


1446 


58 


3 


2770 


4-95 


3865 


.90 


8905 


5.84 


I095 


57 


4 


3067 


4.94 


3811 




92-55 


5.84 


0745 


56 


5 


33 6 3 


4.94 


3757 




9606 


5.83 


0394 


55 


6 


3659 


4-93 


37°4 




9.629956 


5-83 


IO.370044 


54 


7 


3955 


4-93 


3 6 5° 




9.630306 


5.83 


IO.369694 


53 


8 


4251 


4-93 


359 6 




0656 


5.83 


9344 


52 


y 


4547 


4.92 


354^ 




1005 


5.82 


8995 


51 


10 

n 


4842 


4.92 


3488 




1355 


5.82 


8645 


50 
49 


9-595I37 


4.91 


9.963434 


9.631704 


5.82 


10.368296 


12 


543 2 


4.91 


3379 




2°53 


5.81 


7947 


48 


13 


57 2 7 


4.91 


33 2 5 




2401 


5.81 


7599 


47 


14 


6021 


4.90 


3271 




2750 


5.81 


7250 


46 


15 


6315 


4.90 


3217 




3098 


5.80 


6902 


45 


16 


6609 


4.89 


3163 


.90 


3447 


5.80 


6553 


44 


17 


6903 


4.89 


3108 


.91 


3795 


5.80 


6205 


43 


18 


7196 


4.89 


3054 




4H3 


5-79 


5857 


42 


19 


7490 


4.88 


2999 




4490 


5-79 


55io 


41 


20 
21 


7783 


4.88 


2945 




4838 
9.635185 


5-79 


' 5162 


40 
39 


9.598075 


4.87 


9.962890 




5.78 


10.364815 


22 


8368 


4.87 


2836 




553 2 


5.78 


4468 


38 


23 


8660 


4.87 


2781 




5879 


5.78 


4121 


37 


24 


8952 


4.86 


2727 




6226 


5-77 


3774 


36 


25 


9244 


4.86 


2672 




6572 


5-77 


3428 


35 


26 


9536 


4.85 


2617 




6919 


5-77 


3081 


34 


27 


9.599827 


4.85 


2562 




7265 


5-77 


2735 


33 


28 


9.600118 


4.85 


2508 




7611 


5.76 


2389 


32 


29 


0409 


4.84 


2-453 


.91 


7956 


5-76 


2044 


31 


30 
31 


0700 


4.84 


2398 


.92 


8302 


5-76 


1698 


30 
29 


9.600990 


4.84 


9-9 62 343 




9.638647 


5-75 


10.361353 


32 


1280 


4.83 


2288 




8992 


5-75 


1008 


28 


33 


1570 


4.83 


2233 




9337 


5-75 


0663 


27 


34 


i860 


4.82 


2178 




9.639682 


5-74 


10.360318 


26 


36 


2150 


4.82 


2123 




9.640027 


5-74 


i°-359973 


25 


36 


2439 


4.82 


2067 




0371 


5-74 


9629 


24 


37 


2728 


4.81 


2012 




0716 


5-73 


9284 


23 


38 


3017 


4.81 


1957 




1060 


5-73 


8940 


22 


39 


3305 


4.81 


1902 




1404 


5-73 


8596 


21 


40 
41 


3594 


4.80 


1846 




1747 
9.642091 


5-72 


8253 


20 
19 


9.603882 


4.80 


9.961791 




5-72 


io-3579°9 


42 


4170 


4-79 


1735 




2434 


5-72 


7566 


18 


43 


4457 


4-79 


1680 


.92 


2777 


5.72 


7223 


17 


44 


4745 


4-79 


1624 


•93 


3120 


5-7i 


6880 


16 


45 


5032 


4.78 


1569 




34 6 3 


5-7i 


6 537 


15 


46 


53*9 


4.78 


1513 




3806 


5-7i 


6194 


14 


47 


5606 


4.78 


1458 




4148 


5.70 


5852 


13 


48 


5892 


4-77 


1402 




4490 


5.70 


55 10 


12 


49 


6179 


4-77 


1346 




4832 


5-7° 


5168 


11 


50 
51 


6465 


4.76 


1290 




5*74 


5.69 


4826 
10.354484 


10 

9 


9.606751 


4.76 


9.961235 


9.645516 


5.69 


52 


7036 


4.76 


1179 




5857 


5- 6 9 


4H3 


8 


53 


7322 


4-75 


1123 




6199 


5- 6 9 


3801 


7 


54 


7607 


4-75 


1067 




6540 


5.68 


3460 


6 


55 


7892 


4-74 


IOII 




6881 


5.68 


3"9 


5 


56 


8177 


4-74 


0955 




7222 


5.68 


2778 


4 


57 


8461 


4-74 


0899 


•93 


7562 


5- 6 7 


2438 


3 


58 


8745 


4-73 


0843 


•94 


7903 


5- 6 7 


2097 


2 


59 


9029 


4-73 


0786 


•94 


8243 


5- 6 7 


1757 


1 


60 


9.609313 
Cosine. 




9.960730 




9.648583 




10.351417 

Tang. 



M. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1* 


113° 










66° 



65 



24° 




LOGARITHMIC 




155° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 
•94 


Tang. 


Diff. 1" 


Cotang. 


60 


9.609313 


4-73 


9.960730 


9.648583 


5.66 


10.351417 


1 


9597 


4.72 


0674 




8923 


5.66 


1077 


59 


2 


9.609880 


4.72 


0618 




9263 


5.66 


0737 


58 


3 


9.610164 


4.72 


0561 




9602 


5.66 


0398 


57 


4 


0447 


4.71 


°5°5 




9.649942 


5.65 


10.350058 


56 


5 


0729 


4.71 


0448 




9.650281 


5.65 


10.349719 


55 


6 


1012 


4.70 


0392 




0620 


5.65 


9380 


54 


7 


1294 


4.70 


0335 




0959 


5- 6 4 


9041 


53 


8 


1576 


4.70 


0279 




1297 


5.64 


8703 


52 


9 


1858 


4.69 


0222 




1636 


5-64 


8364 


51 


10 
11 


2140 


4.69 


0165 


■94 
•95 


1974 


5-63 


8026 


50 
~49~ 


9.612421 


4.69 


0109 


9.652312 


5- 6 3 


10.347688 


12 


2702 


4.68 


9.960052 




2650 


5-63 


7350 


48 


13 


2983 


4.68 


9-959995 




2988 


5-63 


7012 


47 


14 


3264 


4.67 


9938 




3326 


5.62 


6674 


46 


15 


3545 


4.67 


9882 




3663 


5.62 


6 337 


45 


16 


3^5 


4.67 


9825 




4000 


5.62 


6000 


44 


17 


4105 


4 ;66 


9768 




4337 


5.61 


5663 


43 


18 


4385 


4.66 


9711 




4674 


5.61 


5326 


42 


19 


4665 


4.66 


9654 




5011 


5.61 


4989 


41 


20 
21 


4944 


4.65 


9596 




5348 


5.61 


4652 


40 
39 


9.615223 


4.65 


9-959539 




9.655684 


5.60 


10.344316 


22 


55° 2 


4.65 


9482 




6020 


5.60 


3980 


38 


23 


5781 


4.64 


9425 




6356 


5.60 


3 6 44 


37 


24 


6060 


4.64 


9368 


■95 


6692 


5-59 


33°8 


36 


25 


6338 


4.64 


9310 


.96 


7028 


5-59 


2972 


35 


26 


6616 


4.63 


9 2 53 




73 6 4 


5-59 


2636 


34 


27 


6894 


4- 6 3 


9i95 




7699 


5-59 


2301 


33 


28 


7172 


4.62 


9138 




8034 


5.58 


1966 


32 


29 


7450 


4.62 


9081 




8369 


5.58 


1631 


31 


30 
31 


7727 
9.618004 


4.62 


9023 
9.958965 




8704 


5.58 


1296 


30 
29 


4.61 


9.659039 


5-58 


10.340961 


32 


8281 


4.61 


8908 




9373 


5-57 


0627 


28 


33 


8558 


4.61 


8850 




9.659708 


5-57 


10.340292 


27 


34 


8834 


4.60 


8792 




9.660042 


5-57 


10.339958 


26 


35 


9110 


4.60 


8734 




0376 


5-57 


9624 


25 


36 


9386 


4.60 


8677 




0710 


5.56 


9290 


24 


37 


9662 


4-59 


8619 




i°43 


5.56 


8957 


23 


38 


9.619938 


4-59 


8561 


.96 


1377 


5.56 


8623 


22 


39 


9.620213 


4-59 


8503 


•97 


1710 


5-55 


8290 


21 


40 
41 


0488 


4.58 


8445 




2043 


5-55 


7957 


20 
19 


0763 


4.58 


9.958387 


9.662376 


5-55 


10.337624 


42 


1038 


4-57 


8329 




2709 


5-54 


7291 


18 


43 


1313 


4-57 


8271 




3042 


5-54 


6958 


17 


44 


1587 


4-57 


8213 




3375 


5-54 


6625 


16 


45 


1861 


4.56 


8i54 




37°7 


5-54 


6293 


15 


46 


2135 


4.56 


8096 




4039 


5-53 


5961 


14 


47 


2409 


4.56 


8038 




437 1 


5-53 


5629 


13 


48 


2682 


4-55 


7979 




4703 


5-53 


5297 


12 


49 


2956 


4-55 


7921 




5°35 


5-53 


4965 


11 


50 
51 


3229 
9.623502 


4-55 


7863 
9.957804 




5366 


5-5* 
5-5 2 


4634 


10 
9 


4-54 


•97 


9.665697 


i°-3343°3 


52 


3774 


4-54 


7746 


.98 


6029 


5-5 2 


3971 


8 


53 


4047 


4-54 


7687 




6360 


5-5i 


3640 


7 


54 


43 J 9 


4-53 


7628 




6691 


5-5i 


3309 


6 


55 


459 1 


4-53 


7570 




7021 


5-5i 


2979 


5 


56 


4863 


4-53 


7511 




7352 


5-5i 


2648 


4 


57 


5*35 


4.52 


7452 




7682 


5-5° 


2318 


3 


58 


5406 


4.52 


7393 




8013 


5-5° 


1987 


2 


59 


5677 


4.52 


7335 


.98 


8343 


5-5° 


1657 


1 


60 


9.625948 




9.957276 


Diff.l" 


9.668672 
Cotang. 




10.331328 



M. 


Cosine. 


j Diff. V 


Sine. 


Diff. 1" 


Tang. 


1 114° 










65° 



66 



25° 


SIEVES AND TANGENTS. 


154° 


M. 



Sine. 


Diff. 1" 
4.51 


Cosine. 


Diff.r' 


Tang. 


Diff. 1" 


Cotang. 


60 i 


9.625948 


9.957276 


.98 


9.668673 


5-5° 


10.331327 


J 


6219 


4.51 


7217 




9002 


5-49 


0998 


59 


2 


6490 


4.51 


7158 




9332 


5-49 


0668 


58 


3 


6760 


4.50 


7099 




9661 


5-49 


°339 


57 


4 


7030 


4.50 


7040 




9.669991 


5.48 


10.330009 


56 


5 


7300 


4.50 


6981 


.98 


9.670320 


5.48 


10.329680 


55 


6 


7570 


4.49 


6921 


•99 


0649 


5-48 


935 1 


54 


7 


7840 


4.49 


6862 




0977 


5-48 


9023 


53 


8 


8109 


4.49 


6803 




1306 


5-47 


8694 


52 


9 


8378 


4.48 


6 744 




1634 


5-47 


8366 


51 


10 
11 


8647 
9.628916 


4.48 


6684 
9.956625 




1963 
9.672291 


5-47 


8037 


50 
49 


4-47 




5-47 


10.327709 


12 


9185 


4-47 


6566 




2619 


5-4 6 


738i 


48 


13 


9453 


4-47 


6506 




2947 


5.46 


7053 


47 


14 


9721 


4.46 


6447 




3 2 74 


5.46 


6726 


46 


15 


9.629989 


4.46 


6387 




3602 


5-4 6 


6398 


45 


16 


9.630257 


4.46 


6327 




3929 


5-45 


6071 


44 


17 


0524 


4.46 


6268 


•99 


4257 


5-45 


5743 


43 


18 


0792 


4-45 


6208 


1. 00 


4584 


5-45 


5416 


42 


19 


1059 


4-45 


6148 




4910 


5-44 


5090 


41 


20 
21 


1326 
9- 6 3 I 593 


4-45 


6089 




5 2 37 
9.675564 


5-44 
5-44 


' 4763 
10.324436 


40 
39 


4.44 


9.956029 


22 


1859 


4.44 


5969 




5890 


5-44 


4110 


38 


23 


2125 


4.44 


5909 




6216 


5-43 


3784 


37 


24 


2392 


4-43 


5849 




6543 


5-43 


3457 


36 


25 


2658 


4-43 


5789 




6869 


5-43 


3 1 3 I 


35 


26 


2923 


4-43 


5729 




7194 


5-43 


2806 


34 


27 


3 l8 9 


4.42 


5669 




7520 


5.42 


2480 


33 


28 


3454 


4.42 


5609 




7846 


5.42 


2154 


32 


29 


3719 


4.42 


5548 




8171 


5-4^ 


1829 


31 


30 


3984 


4.41 


5488 


1. 00 

1. 01 


8496 
9.678821 


5.42 


1504 


30 
29 


31 


9.634249 


4.41 


9.955428 


5.41 


10.321179 


32 


45H 


4.40 


5368 




9146 


5.41 


0854 


28 


33 


4778 


4.40 


53°7 




947i 


5-4i 


0529 


27 


34 


5042 


4.40 


5*47 




9.679795 


5.41 


10.320205 


26 


35 


5306 


4-39 


5186 




9.680120 


5.40 


10.319880 


25 


36 


557° 


4-39 


5126 




0444 


5.40 


9556 


24 


37 


5834 


4-39 


5 o6 5 




0768 


5.40 


9232 


23 


38 


6097 


4-39 


5005 




1092 


5.40 


8908 


22 


39 


6360 


4-38 


4944 




1416 


5-39 


8584 


21 


40 
41 


6623 
9.636886 


4.38 


4883 
9.954823 




1740 
9.682063 


5-39 


8260 


20 
19 


4-37 


5-39 


10.317937 


42 


7148 


4-37 


4762 




2387 


5-39 


7613 


18 


43 


741 1 


4-37 


4701 




2710 


5-38 


7290 


17 


44 


7673 


4-37 


4640 




3°33 


5.38 


6967 


16 


4o 


7935 


4.36 


4579 


I.OI 


335 6 


5.38 


6644 


15 


46 


8197 


4-3 6 


4518 


1.02 


3679 


5.38 


6321 


14 


47 


8458 


4-3 6 


4457 




4001 


5-37 


5999 


13 


48 


8720 


4-35 


439 6 




4324 


5-37 


5676 


12 


49 


8981 


4-35 


4335 




4646 


5-37 


5354 


11 


50 
51 


9242 
9503 


4-35 


4274 
9.954213 




4968 


5-37 


5032 
10.314710 


10 
9 


4-34 




9.685290 


5-3 6 


52 


9.639764 


4-34 


4i5 2 




5612 


5-3 6 


4388 


8 


53 


9.640024 


4-34 


4090 




5934 


5-3 6 


4066 


7 


54 


0284 


4-33 


4029 




6255 


5-3 6 


3745 


6 


55 


0544 


4-33 


3968 




6577 


5-35 


34*3 


5 


56 


0804 


4-33 


3906 




6898 


5-35 


3102 


4 


57 


1064 


4-3 2 


3845 




7219 


5-35 


2781 


3 


58 


I3 2 4 


4.32 


3783 


1.02 


754o 


5-35 


2460 


2 


59 


1583 


4.32 


3722 


1.03 


7861 


5-34 


2139 


1 


60 


9.641842 
Cosine. 




9.953660 


Diff.l" 


9.688182 
Cotang. 




10.311818 




M. 


Diff. 1" 


Sine. 


Diff. 1" 


Tang. 


115° 








04° 



b7 



26° 




LOGARITHMIC 




153° 




M. 




Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 




9.641842 


4-3 1 


9.953660 


I.03 


9.688182 


5-34 


IO.311818 




1 


2101 


4.31 


3599 




8502 


5-34 


1498 


59 




2 


2360 


4-31 


3537 




8823 


5-34 


1177 


58 




3 


2618 


4.30 


3475 




9H3 


5-33 


0857 


57 




4 


2877 


4-3° 


3413 




9463 


5-33 


0537 


56 




5 


3 J 35 


4-3° 


3352 




9.689783 


5-33 


IO.310217 


65 




6 


3393 


4-3° 


3290 




9.690103 


5-33 


10.309897 


54 




7 


3650 


4.29 


3228 




0423 


5-33 


9577 


53 




8 


3908 


4.29 


3166 




0742 


5-32 


9258 


52 




9 


4165 


4.29 


3104 




1062 


5-3 2 


8938 


61 




10 
11 


4423 


4.28 


3042 


I.03 
I.04 


1381 


5-3 2 


8619 


50 

49 




9.644680 


4.28 


9.952980 


9.691700 


5-3 1 


10.308300 




12 


4936 


4.28 


2918 




2019 


5-3 1 


7981 


48 




13 


5193 


4.27 


2855 




2338 


5-3i 


7662 


47 




14 


545° 


4.27 


2793 




2656 


5-3i 


7344 


46 




15 


5706 


4.27 


2731 




2975 


5-3i 


7025 


45 




16 


5962 


4.26 


2669 




3 2 93 


5-3° 


6707 


44 




17 


6218 


4.26 


2606 




3612 


5-3° 


6388 


43 




18 


6474 


4.26 


2544 




393° 


5-3° 


6070 


42 




19 


6729 


4.26 


2481 




4248 


5-3° 


575* 


41 




20 
21 


6984 


4.25 


2419 




4566 


5- 2 9 


5434 


40 
39 




9.647240 


4.25 


9.952356 




9.694883 


5- 2 9 


10.305 1 17 




22 


7494 


4.24 


2294 




5201 


5.29 


4799 


. 38 




23 


7749 


4.24 


2231 


I.04 


55i8 


5- 2 9 


4482 


37 




24 


8004 


4.24 


2168 


I.05 


5836 


5- 2 9 


4164 


36 




25 


8258 


4.24 


2106 




6i53 


5.28 


3847 


35 




26 


8512 


4-^3 


2043 




6470 


5.28 


353° 


34 




27 


8766 


4- 2 3 


1980 




6787 


5.28 


3 2I 3 


33 




28 


9020 


4.23 


1917 




7103 


5.28 


2897 


32 




29 


9274 


4.22 


1854 




7420 


5.27 


2580 


31 




30 
31 


9527 


4.22 


1791 




773 6 
9.698053 


5-^7 


2264 


30 
29 




9.649781 


4.22 


9.951728 




5-*7 


10.301947 




32 


9.650034 


4.22 


1665 




8369 


5.27 


1631 


28 




33 


0287 


4.21 


1602 




8685 


5.26 


I3!5 


27 




34 


0539 


4.21 


1539 




9001 


5.26 


0999 


26 




35 


0792 


4.21 


1476 




9316 


5.26 


0684 


25 




36 


1044 


4.20 


1412 


I.05 


9632 


5.26 


0368 


24 




37 


1297 


4.20 


1349 


I.06 


9.699947 


5.26 


10.300053 


23 




38 


1549 


4.20 


1286 




9.700263 


5.25 


10.299737 


22 




39 


1800 


4.19 


1222 




0578 


5- 2 5 


9422 


21 




40 
41 


2052 


4.19 


1159 




0893 


5.25 


9107 


20 
19 




9.652304 


4.19 


9.951096 




9.701208 


5.24 


10.298792 




42 


2555 


4.18 


1032 




1523 


5- 2 4 


8477 


18 




43 


2806 


4.18 


0968 




1837 


5.24 


8163 


17 




44 


3057 


4.18 


0905 




2152 


5.24 


7848 


16 




45 


3308 


4.18 


0841 




2466 


5.24 


7534 


15 




46 


3558 


4.17 


0778 




2780 


5- 2 3 


7220 


14 




47 


3808 


4.17 


0714 




3°95 


5- 2 3 


6905 


13 




48 


4059 


4.17 


0650 




34°9 


5- 2 3 


6591 


12 




49 


4309 


4.16 


0586 


I.06 


3723 


5- 2 3 


6277 


11 




50 


4558 


4.16 


0522 


I.07 


4036 


5.22 
5.22 


5964 


10 
9 




51 


9.654808 


4.16 


9.950458 




9.704350 


10.295650 




52 


5058 


4.16 


0394 




4663 


5.22 


5337 


8 




53 


53°7 


4.15 


0330 




4977 


5.22 


5023 


7 




54 


5556 


4.15 


0266 




5290 


5.22 


4710 


6 




55 


5805 


4.15 


0202 




5603 


5.21 


4397 


6 




56 


6054 


4.14 


0138 




5916 


5.21 


4084 


4 




57 


6302 


4.14 


0074 




6228 


5.21 


3772 


3 




58 


655i 


4.14 


9.950010 




6541 


5.21 


3459 


2 




59 


6799 


4.13 


9.949945 


I.07 


6854 


5.21 


3146 


1 




60 


9.657047 
Cosine. 


Diff. 1" 


9.949881 




9.707166 




10.292834 




M. 




Sine. 


Diff.]" 


Cotang. 


Diff. 1" 


Tang. 




116° 








63° 





08 



27° 


SIKTES AND TANCfrENTS. 


152° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 
9.707166 


Diff. 1" 


Cotang. 


l 

60 


9.657047 


4.13 


9.949881 


1.07 


5.20 


10.292834 


1 


7295 


4-13 


9816 


I.07 


7478 


5- 


20 


2522 


59 


2 


754 2 


4.12 


9752 


I.07 


7790 


5 


20 


2210 


58 


8 


7790 


4.12 


9688 


I.08 


8102 


5 


20 


1898 


57 


: 4 


8037 


4.12 


9623 




8414 


5 


19 


1586 


56 





8284 


4.12 


9558 




8726 


5 


19 


1274 


55 


6 


853i 


4.1 1 


9494 




9037 


5 


19 


0963 


54 


7 


8778 


4.11 


9429 




9349 


5 


19 


0651 


53 


8 


9025 


4.1 1 


9364 




9660 


5 


19 


0340 


52 


y 


9271 


4.10 


9300 




9.709971 


5 


18 


IO.290029 


51 


10 

n 


9517 


4.10 


9235 




9.710282 
°593 


5 


18 


10.289718 


50 
49 


9.659763 


4.10 


9.949170 




5 


18 


9407 


12 


9.660009 


4.09 


9105 




0904 


5 


18 


9096 


48 


13 


0255 


4.09 


9040 




1215 


5 


18 


8785 


47 


14 


0501 


4.09 


8975 




1525 


5 


17 


8475 


46 


15 


0746 


4.09 


8910 




1836 


5 


17 


8164 


45 


16 


0991 


4.08 


8845 


I.08 


2146 


5 


17 


7854 


44 


17 


1236 


4.08 


8780 


I.09 


2456 


5 


17 


7544 


43 


18 


1481 


4.08 


8715 




2766 


5 


16 


7234 


42 


19 


1726 


4.07 


8650 




3076 


5 


16 


6924 


41 


20 
21 


1970 
9.662214 


4.07 


8584 




3386 


5 


16 


6614 


40 
39 


4.07 


9.948519 




9.713696 


5 


16 


10.286304 


22 


2459 


4.07 


8454 




4005 


5 


16 


5995 


38 


23 


2703 


4.06 


8388 




4314 


5 


M 


5686 


37 


24 


2946 


4.06 


8323 




4624 


5 


15 


5376 


36 


25 


3190 


4.06 


8257 




4933 


5 


15 


5067 


35 


26 


3433 


4.05 


8192 




5242 


5 


!5 


4758 


34 


27 


3 6 77 


4.05 


8126 




555i 


5 


J 4 


4449 


33 


28 


3920 


4.05 


8060 


I.09 


5860 


5 


H 


4140 


32 


29 


4163 


4.05 


7995 


I. IO 


6168 


5 


x 4 


3832 


31 


30 
31 


4406 


4.04 


7929 




6477 


5 


H 


35^3 


30 
29 


9.664648 


4.04 


9.947863 




9.716785 


5 


14 


10.283215 


32 


4891 


4.04 


7797 




7093 


5 


13 


2907 


28 


33 


5133 


4-°3 


7731 




7401 


5 


!3 


2599 


27 


34 


5375 


4.03 


7665 




7709 


5 


J 3 


2291 


26 


35 


5 6 i7 


4.03 


7600 




8017 


5 


J 3 


1983 


25 


36 


5859 


4.02 


7533 




8325 


5 


13 


1675 


24 


37 


6100 


4.02 


7467 




8633 


5 


12 


1367 


23 


38 


6342 


4.02 


7401 




8940 


5 


.12 


1060 


22 


39 


6583 


4.02 


7335 




9248 


5 


.12 


0752 


21 


40 
41 


6824 


4.01 
4.01 


7269 




9555 


5 


.12 


0445 


20 
19 


9.667065 


9.947203 


I. IO 


9.719862 


5 


.12 


10.280138 


42 


73°5 


4.01 


7136 


I. II 


9.720169 


5 


11 


10.279831 


18 


43 


7546 


4.01 


7070 




0476 


5 


.11 


9524 


17 


44 


7786 


4.00 


7004 




0783 


5 


11 


9217 


16 


45 


8027 


4.00 


6937 




1089 


5 


.11 


8911 


15 


46 


8267 


4.00 


6871 




1396 


5 


.11 


8604 


14 


47 


8506 


3-99 


6804 




1702 


5 


IO 


8298 


13 


48 


8746 


3-99 


6738 




2009 


5 


IO 


7991 


12 


49 


8986 


3-99 


6671 




2315 


5 


IO 


7685 


11 


50 
51 


9225 


3-99 


6604 




2621 


5 


IO 


7379 
10.277073 


10 
9 


9464 


3.98 


9.946538 


9.722927 


5 


.10 


52 


9703 


3.98 


6471 




3232 


5 


09 


6768 


8 


53 


9.669942 


3-98 


6404 




3538 


5 


.09 


6462 


7 


54 


9.670181 


3-97 


6337 


I. II 


3844 


5 


09 


6156 


6 


55 


0419 


3-97 


6270 


1. 12 


4149 


5 


.09 


5851 


5 


56 


0658 


3-97 


6203 




4454 


5 


.09 


5546 


4 


57 


0896 


3-97 


6136 




4759 


5 


.08 


5241 


3 


58 


"34 


3.96 


6069 




5065 


5 


.08 


4935 


2 


59 


1372 


3.96 


6002 


1. 12 


5369 


5 


.08 


4631 


1 


60 


9.671609 




9-945935 




9.725674 




10.274326 




M. ! 


Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1" 


Tang. 


117° 








62° | 



26 



69 



28° 




LOGARITHMIC 




151° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.671609 


3-9 6 


9-945935 


1. 12 


9.725674 


5.08 


IO.274326 


1 


1847 


3-95 


5868 




5979 


5.08 


4021 


59 


2 


2084 


3-95 


5800 




6284 


5-°7 


3716 


58 


3 


2321 


3-95 


5733 




6588 


5.07 


3412 


57 


4 


2 55 8 


3-95 


5666 




6892 


5.07 


3108 


56 


5 


2795 


3-94 


5598 




7197 


5.07 


2803 


55 


6 


3°3 2 


3-94 


553i 


1. 12 


75oi 


5.07 


2499 


54 


7 


3268 


3-94 


5464 


1. 13 


7805 


5.06 


2195 


53 


1 8 


35°5 


3-94 


539 6 




8109 


5.06 


1891 


52 


9 


374i 


3-93 


53*8 




8412 


5.06 


1588 


51 


10 
11 


3977 
9.674213 


3-93 


5261 




8716 


5.06 


1284 


50 
49 


3-93 


9-945I93 




9.729020 


5.06 


IO.270980 


12 


4448 


3.92 


5125 




93*3 


5-05 


0677 


48 


13 


4684 


3-9 2 


5058 




9626 


5-°5 


0374 


47 


14 


4919 


3-9 2 


4990 




9.729929 


5-°5 


IO.270071 


46 


15 


5 J 55 


3-9 2 


4922 




9-73° 2 33 


5-o5 


IO.269767 


45 


16 


539° 


3-9 1 


4854 




0535 


5-°5 


9465 


44 


17 


5624 


3-9 1 


4786 




0838 


5.04 


9162 


43 


18 


5859 


3-9 1 


4718 




1141 


5-°4 


8859 


42 


19 


6094 


3-9 1 


4650 


113 


1444 


5-°4 


8556 


41 


20 
21 


6328 


3-9° 


4582 


1. 14 


1746 


5-o4 


8254 


40 
39 


9.676562 


3.90 


9.944514 


9.732048 


5.04 


IO.267952 


22 


6796 


3.90 


4446 




2351 


5-°3 


7649 


38 


23 


7030 


3-9° 


4377 




2653 


5-°3 


7347 


37 


24 


7264 


3.89 


4309 




2955 


5-°3 


7045 


36 


25 


7498 


3-89 


4241 




3^57 


5-^3 


6743 


35 


26 


773 1 


3.89 


4172 




3558 


5-°3 


6442 


34 


27 


7964 


3.88 


4104 




3860 


5.02 


6140 


33 


28 


8197 


3.88 


4036 




4162 


5.02 


5838 


32 


29 


8430 


3.88 


3967 




44 6 3 


5.02 


5537 


31 


30 
31 


8663 


3.88 


3899 




4764 


5.02 


5236 


30 
29 


9.678895 


3.87 


9.943830 




9.735066 


5.02 


10.264934 


32 


9128 


3.87 


3761 


1. 14 


53 6 7 


5.02 


4 6 33 


28 


33 


9360 


3.87 


3 6 93 


1. 15 


5668 


5.01 


4332 


27 


34 


9592 


3-87 


3624 




59 6 9 


5.01 


4031 


26 


35 


9.679824 


3.86 


3555 




6269 


5.01 


373 1 


25 


36 


9.680056 


*!£ 


3486 




6570 


5.01 


343° 


24 j 


37 


0288 


3.86 


34i7 




6871 


5.01 


3129 


23 I 


38 


0519 


3.85 


3348 




7171 


5.00 


2829 


22 


39 


0750 


3-85 


3*79 




7471 


5.00 


2529 


21 


40 
41 


0982 


3-85 
3-85 


3210 




7771 


5.00 


2229 


20 
19 


9.681213 


9.943141 




9.738071 


5.00 


10.261929 


42 


1443 


3-84 


3072 




8371 


5.00 


1629 


18 


43 


1674 


3.84 


3003 




8671 


4.99 


1329 


17 


44 


1905 


3-84 


2934 




8971 


4.99 


1029 


16 


45 


2135 


3-84 


2864 


1. 15 


9271 


4.99 


0729 


15 


46 


2365 


3.83 


2795 


1. 16 


9570 


4.99 


0430 


14 


47 


259S 


3-83 


2726 




9.739870 


4.99 


10.260130 


13 


48 


2825 


3-83 


2656 




9.740169 


4.99 


10.259831 


12 


49 


3°55 


3-83 


2587 




0468 


4.98 


9532 


11 


50 
51 


3284 
9.683514 


3.82 


2517 




0767 


4.98 


9233 


10 
9 


3.82 


9.942448 




9.741066 


4.98 


10.258934 


52 


3743 


3.82 


2378 




i3 6 5 


4.98 


863s 


8 


53 


3972 


3.82 


2308 




1664 


4.98 


8336 


7 


54 


4201 


3.81 


2239 




1962 


4-97 


8038 


6 


55 


443° 


3.81 


2169 




2261 


4-97 


7739 


5 


56 


4658 


3.81 


2099 




2559 


4-97 


7441 


4 


57 


4887 


3.80 


2029 




2858 


4-97 


7142 


3 


58 


5H5 


3.80 


1959 


1. 16 


3 J 5 6 


4-97 


6844 


2 


59 


™ 5343 


3.80 


1889 


1. 17 


3454 


4-97 


6546 


1 


60 


9.685571 
Cosine. 




9.941819 


Diff.l" 


9-743752 
Cotang. 




10.256248 





Diff. 1" 


Sine. 


Diff. l" 


Tang. 


M. 


118° 








61° 



70 



29° 


SINES AND TANGENTS. 


150° 


M. 




Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.685571 


3.80 


9.941819 


1. 17 


9-74375 2 


4.96 


IO.256248 


1 


5799 


3 


79 


1749 




4050 


4.96 


595° 


59 


2 


6027 


3 


79 


1679 




4348 


4.96 


5 6 5 2 


58 


3 


6254 


3 


79 


1609 




4645 


4.96 


5355 


57 


4 


6482 


3 


79 


1539 




4943 


4.96 


5°57 


56 


5 


6709 


3 


78 


1469 




5 2 4° 


4-95 


4760 


55 


6 


6936 


3 


78 


1398 




5538 


4-95 


4462 


54 


7 


7163 


3 


78 


1328 




5835 


4-95 


4165 


53 


8 


73 8 9 


3 


78 


1258 




6132 


4-95 


3868 


52 


9 


7616 


3 


77 


1187 




6429 


4-95 


357i 


51 


10 

11 


7843 


3 


77 


1117 


I.17 

1. 18 


6726 


4-95 


3274 


50 
49 


9.688069 


3 


77 


9.941046 


9.747023 


4.94 


10.252977 


12 


8295 


3 


77 


0975 




7319 


4.94 


2681 


48 


13 


8521 


3 


76 


0905 




7616 


, 4-94 


2384 


47 


14 


8747 


3 


76 


0834 




79*3 


4.94 


2087 


46 


15 


8972 


3 


76 


0763 




8209 


4.94 


1791 


45 


16 


9198 


3 


76 


0693 




8505 


4-93 


1495 


44 


IV 


9423 


3 


75 


0622 




8801 


4-93 


1199 


43 


18 


9648 


3 


75 


0551 




9097 


4-93 


0903 


42 


19 


9.689873 


3 


75 


0480 




9393 


4-93 


0607 


41 


20 
21 


9.690098 
0323 


3 


75 


0409 




9689 
9.749985 


4-93 
4-93 


03 1 1 


40 
39 


3 


74 


9.940338 


10.250015 


22 


0548 


3 


74 


0267 




9.750281 


4-93 


10.249719 


38 


23 


0772 


3 


74 


OI96 


1. 18 


0576 


4.92 


9424 


37 


24 


0996 


3 


74 


0125 


1. 19 


0872 


4.92 


9128 


36 


25 


1220 


3 


73 


9.940054 




1167 


4.92 


8833 


35 


26 


1444 


3 


73 


9.939982 




1462 


4.92 


8538 


34 


27 


1668 


3 


73 


9911 




1757 


4.92 


8243 


33 


28 


. 1892 


3 


73 


9840 




2052 


4.91 


7948 


32 


29 


2115 


3 


72 


9768 




2347 


4.91 


7653 


31 


30 
31 


2339 
9.692562 


3 


72 


9697 
9.939625 




2642 
9-75 2 937 


4.91 


7358 
10.247063 


30 
29 


3 


72 


4.91 


32 


2785 


3 


7i 


9554 




3 2 3 J 


4.91 


6769 


28 


33 


3008 


3 


7i 


9482 




3526 


4.91 


6474 


27 


34 


3 2 3 J 


3 


7i 


9410 




3820 


4.90 


6180 


26 


35 


3453 


3 


7i 


9339 


1. 1 9 


4115 


4.90 


5885 


25 


36 


3676 


3 


70 


9267 


I.20 


4409 


4.90 


559 1 


24 


3V 


3898 


3 


70 


9 J 95 




4703 


4.90 


5297 


23 


38 


4120 


3 


70 


9123 




4997 


4.90 


5003 


22 


39 


4342 


3 


70 


9052 




5 2 9! 


4.90 


4709 


21 


40 
41 


4564 
9.694786 


3 


69 


8980 
9.938908 




5585 
9.755878 


4.89 
4.89 


4415 


20 

~i<r 


3 


69 


10.244122 


42 


5007 


3 


6 9 


8836 




6172 


4.89 


3828 


18 


43 


5229 


3 


6 9 


8763 




6465 


4.89 


3535 


17 


44 


545o 


3 


68 


8691 




6759 


4.89 


3 2 4i 


16 


45 


5 6 7i 


3 


68 


8619 




7052 


4.89 


2948 


15 


46 


5892 


3 


68 


8547 




7345 


4.88 


2655 


14 


47 


6113 


3 


68 


8475 


I.20 


7638 


4.88 


2362 


13 


48 


6 334 


3 


67 


8402 


1. 21 


7931 


4.88 


2069 


12 


49 


6 554 


3 


67 


8330 




8224 


4.88 


1776 


11 


50 
51 


6775 
9.696995 


3 


67 


8258 
9.938185 




8517 
9.758810 


4.88 


1483 


10 
9 


3 


67 


4.88 


10.241190 


52 


7215 


3 


66 


8113 




9102 


4.87 


0898 


8 


53 


7435 


3 


66 


8040 




9395 


4.87 


0605 


7 


54 


7654 


3 


66 


7967 




9687 


4.87 


0313 


6 


55 


7874 


3 


66 


7895 




9-759979 


4.87 


10.240021 


5 


56 


8094 


3 


65 


7822 




9.760272 


4.87 


10.239728 


4 


57 


8313 


3 


65 


7749 




0564 


4.87 


9436 





58 


8532 


3 


65 


7676 




0856 


4.86 


9144 


2 


59 


8751 


3 


65 


7604 


1. 21 


1148 


4.86 


8852 


1 


60 


9.698970 
Cosine. 




9-937531 
Sine. 


Diff.l" 


9.761439 


Diff. 1" 


10.238561 
Tang. 



M. 


Diff. 1" 


Cotang. 


119° 








60° 



30° 




IiOGAXUTHXVXXC 




149° 


M. 

~~ 


Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.698970 


3- 6 4 


9-937531 


1. 21 


9.761439 


4.86 


10.238561 


1 


9189 


3- 6 4 


7458 


1.22 


I73 1 


4.86 


8269 


59 


2 


9407 


3- 6 4 


7385 




2023 


4.86 


7977 


58 


3 


9626 


3- 6 4 


7312 




2314 


4.86 


7686 


57 


4 


9.699844 


3.63 


7238 




2606 


4.85 


7394 


56 


5 


9.700062 


3- 6 3 


7165 




2897 


4.85 


7103 


55 


6 


0280 


3- 6 3 


7092 




3188 


4.85 


6812 


54 


7 


0498 


3.63 


7019 




3479 


4.85 


6521 


53 


8 


0716 


3- 6 3 


6946 




377o 


4.85 


6230 


52 


9 


0933 


3.62 


6872 




4061 


4.85 


5939 


51 


10 

11 


IISI 

9.701368 


3.62 


6799 


1.22 


4352 


4.84 


5648 


50 
49 


3.62 


9.936725 


9.764643 


4.84 


10.235357 


12 


1585 


3.62 


6652 


I.23 


4933 


4.84 


5067 


48 


13 


1802 


3- 6 . 1 


6578 




5224 


4.84 


4776 


47 


14 


2019 


3.61 


65 5 




55H 


4.84 


4486 


46 


15 


2*136 


3.61 


6431 




5805 


4.84 


4*95 


45 


16 


2452 


3.61 


6357 




6095 


4.84 


39°5 


44 


17 


2669 


3.60 


6284 




6385 


4-83 


3 6l 5 


43 


18 


2885 


3.60 


6210 




6675 


4.83 


33^5 


42 


19 


3IOI 


3.60 


6136 




6965 


4.83 


3°35 


41 


20 
21 


33*7 


3.60 


6062 




7255 


4.83 


2745 


40 
39 


9-7°3533 


3-59 


9'9359 8 8 




9.767545 


4-83 


10.232455 


22 


3749 


3-59 


59H 




7834 


4-83 


2166 


38 


28 


3964 


3-59 


5840 


I.23 


8124 


4.82 


1876 


37 


24 


4179 


3-59 


5766 


I.24 


8413 


4.82 


1587 


36 


25 


4395 


3-59 


5692 




8703 


4.82 


1297 


35 


28 


4610 


3-58 


56l8 




8992 


4.82 


1008 


34 


27 


4825 


3-58 


5543 




9281 


4.82 


0719 


33 


28 


5040 


3-58 


5469 




9570 


4.82 


0430 


32 


29 


5^54 


3-58 


5395 




9.769860 


4.81 


10.230140 


31 


30 
31 


5469 


3-57 


5320 




9.770148 


4.81 


10.229852 


30 
29 


9.705683 


3-57 


9.935246 




0437 


4.81 


9563 


32 


5898 


3-57 


5171 




0726 


4.81 


9274 


28 


33 


6112 


3-57 


5°97 




1015 


4.81 


8985 


27 


34 


6326 


3-5 6 


5022 




1303 


4.81 


8697 


26 


35 


6539 


3-5 6 


4948 




1592 


4.81 


8408 


25 


36 


6753 


3-5 6 


4873 


I.24 


1880 


4.80 


8120 


24 


37 


6967 


3-5 6 


4798 


I.25 


2168 


4.80 


7832 


23 


38 


7180 


3-55 


4723 




2457 


4.80 


7543 


22 


39 


7393 


3-55 


4649 




2 745 


4.80 


7255 


21 


40 
41 


7606 
9.707819 


3-55 


4574 




3°33 


4.80 


6967 


20 
19 


3-55 


9.934499 




9.773321 


4.80 


10.226679 


42 


8032 


3-54 


4424 




3608 


4-79 


6392 


18 


43 


8245 


3-54 


4349 




3896 


4-79 


6104 


17 


44 


8458 


3-54 


4 2 74 




4184 


4-79 


5816 


16 


45 


8670 


3-54 


4199 




447i 


4-79 


5529 


15 


46 


8882 


3-53 


4123 




4759 


4-79 


5 2 4i 


14 


47 


9094 


3-53 


4048 




5046 


4-79 


4954 


13 


48 


9306 


3-53 


3973 


I.25 


5333 


4-79 


4667 


12 


49 


9518 


3-53 


3898 


I.26 


5621 


4.78 


4379 


11 


50 
51' 


9730 


3-53 


3822 




5908 


4.78 
4.78 


4092 
10.223805 


10 
9 


9.709941 


3-52 


9-933747 


9.776195 


52 


9.710153 


3-5* 


3671 




6482 


4.78 


35i8 


8 


53 


0364 


3-52 


3596 




6769 


4.78 


3 2 3J 


7 


54 


o575 


3-5 2 


- 352o 




7055 


4.78 


2945 


6 


55 


0786 


3-5i 


3445 




7342 


4.78 


2658 


5 


56 


0997 


3-5i 


33 6 9 




7628 


4-77 


2372 


4 


57 


1208 


3-5i 


3 2 93 




79*5 


4-77 


2085 


3 


58 


1419 


3-5i 


3217 




8201 


4-77 


1799 


2 


59 


1629 


3-5° 


3 J 4i 


I.26 


8487 


4-77 


1513 


1 


60 


9.711839 




9.933066 




9.778774 




10.221226 




M. 


Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1" 


Tang. 


120° 
t — — 








59° 



72 



31° 


SINES AND TANGENTS. 


148° 


M. 



Sine. 


Diflf. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.711839 


3.50 


9.933066 


I.26 


9.778774 


4-77 


IO.221226 


1 


2050 


3-5° 


2990 


I.27 


9060 


4-77 


0940 


59 


2 


2260 


3-5° 


2914 




9346 


4.76 


0654 


58 


3 


2469 


3-49 


2838 




9632 


4.76 


0368 


57 


4 


2679 


3-49 


2762 




9.779918 


4.76 


10.220082 


56 


5 


2889 


3-49 


2685 




9.780203 


4.76 


10.219797 


55 


6 


3098 


3-49 


2609 




0489 


4.76 


9511 


54 


1 


3308 


3-49 


2533 




°775 


4.76 


9225 


53 


8 


35 J 7 


348 


M57 




1060 


4.76 


8940 


52 


9 


3726 


3.48 


2380 




1346 


4-75 


8654 


51 1 


10 
11 


3935 


3.48 


2304 




1631 


4-75 


8369 
10.218084 


50 

49 


9.714144 


3-48 


9.932228 




9.781916 


4-75 


12 


435 2 


3-47 


2151 


I.27 


2201 


4-75 


7799 


48 


13 


4561 


3-47 


2075 


I.28 


2486 


4-75 


7SH 


47 


14 


4769 


3-47 


1998 




2771 


4-75 


7229 


46 


15 


4978 


3-47 


1921 




3056 


4-75 


6944 


45 


16 


5186 


3-47 


1845 




334i 


4-75 


6659 


44 


17 


5394 


3.46 


I768 




3626 


4-74 


6374 


43 


18 


5602 


34 6 


1691 




3910 


4-74 


6090 


42 


19 


5809 


3-46 


1614 




4195 


4-74 


, 5805 


41 


20 
21 


6017 
9.716224 


3.46 


1537 




4479 


4-74 
4-74 


5521 


40 
~89~ 


3-45 


9.931460 




9.784764 


10.215236 


22 


6432 


3-45 


1383 




5048 


4-74 


4952 


3S 


23 


6639 


3-45 


1306 


I.28 


533* 


4-73 


4668 


37 


24 


6846 


3-45 


1229 


I.29 


5616 


4-73 


4384 


36 


25 


7053 


3-45 


II52 




5900 


4-73 


4100 


35 


26 


7259 


3-44 


I075 




6184 


4-73 


3816 


34 


27 


7466 


3-44 


0998 




6468 


4-73 


3532- 


33 


28 


7673 


3-44 


0921 




6752 


4-73 


3248 


32 


29 


7879 


3-44 


0843 




7036 


4-73 


2964 


31 


30 
31 


8085 


3-43 


0766 




7319 


4.72 


2681 


30 

29 


9.718291 


3-43 


9.930688 




9.787603 


4.72 


10.212397 


32 


8497 


3-43 


06 1 1 




7886 


4.72 


2114 


28 


33 


8703 


3-43 


°533 




8170 


4.72 


1830 


27 


34 


8909 


3-43 


0456 




8453 


4.72 


1547 


26 


35 


9114 


3-42 


0378 


I.29 


8736 


4.72 


1264 


25 


36 


9320 


3-42 


0300 


I.30 


9019 


4.72 


0981 


24 


37 


95^5 


3-4^ 


0223 




9302 


4.71 


0698 


23 


38 


973° 


3.42 


0145 




9585 


4.71 


0415 


22 


39 


9-7I9935 


3-4i 


9.930067 




9.789868 


4.71 


10.210132 


21 


40 
41 


9.720140 


34i 

3-4i 


9.929989 




9.790151 
0433 


4.71 
4.71 


10.209849 
9567 


20 
19 


0345 


9911 


42 


0549 


34 1 


9833 




0716 


4.71 


9284 


18 


43 


0754 


3-4° 


9755 




0999 


4.71 


9001 


17 


44 


0958 


3.40 


9677 




1281 


4.71 


8719 


16 


45 


1162 


3-4° 


9599 




1563 


4.70 


8437 


15 


46 


1366 


3-4° 


9521 




1846 


4.70 


8154 


14 


47 


1570 


3-4° 


9442 


I.30 


2128 


4.70 


7872 


13 


48 


1774 


3-39 


93 6 4 


1. 31 


2410 


4.70 


7590 


12 


49 


1978 


3-39 


9286 




2692 


4.70 


7308 


11 


50 
51 


2181 


3-39 
3-39 


9207 




2974 
9.793256 


4.70 
4.70 


7026 


10 
' 9 


9.722385 


9.929129 


10.206744 


52 


2588 


3-39 


9050 




3538 


4.69 


6462 


S 


53 


2791 


3-38 


8972 




3819 


4.69 


6181 


1 


54 


2994 


3-38 


8893 




4101 


4.69 


5899 


6 


55 


3*97 


3-38 


8815 




4383 


4.69 


5617 


5 


56 


3400 


3-38 


8736 




4664 


4.69 


533 6 


4, 


57 


3603 


3-37 


8657 




4945 


4.69 


5°55 


* 1 


58 


3805 


3-37 


8578 




5227 


4.69 


4773 


? i 


59 


4007 


3-37 


8499 


1. 31 


55o8 


4.68 


4492 


1 ; 


; GO 

i 


9.724210 




9.928420 

Sine. 


Diff.l" 


9.795789 
Cotang. 


Diff. 1" 


10.20421 1 

Tang. 


| 

M. j 


Cosine. 


Diff. 1" 


121° 








58° 



io 



.—■— — — .. 
32° 




LOGARITHMIC 




147° 




M. 




Sine. 


Diff. 1" 


Cosine. 


Diff.l' 
I.32 


Tan-. 


Diff. 1" 


Cotang. 


60 




9.724210 


3-37 


9.928420 


9.795789 


4.68 


IO.20421 1 




1 


4412 


3-37 


8342 




6070 


4.68 


393° 


59 




2 


4614 


3-3 6 


8263 




6 35I 


4.68 


3649 


58 




3 


4816 


3-36 


8183 




6632 


4.68 


3368 


57 




4 


5017 


3-36 


8104 




6913 


4.68 


3087 


56 




5 


5 2 *9 


3-3 6 


8025 




7194 


4.68 


2806 


55 




6 


5420 


3-35 


7946 




7475 


4.68 


2525 


54 




7 


5622 


3-35 


7867 




7755 


4.68 


2245 


53 




8 


5^3 


3-35 


7787 




8036 


4.67 


1964 


52 




9 


6024 


3-35 


7708 




8316 


4.67 


1684 


51 




10 
11 


6225 


3-35 


7629 




8596 


4.67 


1404 


50 
49 




9.726426 


3-34 


9.927549 


I.32 


9.798877 


4.67 


10.201123 




12 


6626 


3-34 


7470 


I.33 


9157 


4.67 


0843 


48 




13 


6827 


3-34 


739° 




9437 


4.67 


0563 


47 




14 


7027 


3-34 


7310 




9717 


4.67 


0283 


46 




15 


7228 


3-34 


7231 




9.799997 


4.66 


10.200003 


45 




16 


7428 


3-33 


7151 




9.800277 


4.66 


10.199723 


44 




17 


7628 


3-33 


7071 




o557 


4.66 


9443 


43 




18 


7828 


3-33 


6991 




0836 


4.66 


9164 


42 




19 


8027 


3-33 


6911 




1116 


4.66 


8884 


41 




20 
21 


8227 


3-33 


6831 




1396 


4.66 


8604 


40 
39 




9.728427 


3-3 2 


9.926751 




9.801675 


4.66 


10.198325 




22 


8626 


3-3 2 


6671 




1955 


4.66 


8045 


38 




23 


8825 


3-3 2 


6591 


I.33 


2234 


4.65 


7766 


37 




24 


9024 


3-3 2 


6511 


I.34 


2 5*3 


4.65 


7487 


36 




25 


9223 


3-3 1 


6431 




2792 


4- 6 5 


7208 


35 




26 


9422 


3-3 1 


6 35i 




3072 


4.65 


6928 


34 




27 


9621 


3-3 1 


6270 




335i 


4.65 


6649 


33 




28 


9.729820 


3-3i 


6190 




3630 


4.65 


6370 


32 




29 


9.730018 


3-3° 


6110 




3908 


4.65 


6092 


31 




30 
31 


0216 


3-3° 


6029 




4187 


4.65 


5813 


30 
29 




0415 


3-3° 


9.925949 




9.804466 


4.64 


10.195534 




32 


0613 


3-3° 


5868 




4745 


4.64 


5 2 55 


28 




33 


0811 


3-3° 


5788 




5023 


4.64 


4977 


27 




34 


IO09 


3- 2 9 


5707 




53° 2 


4.64 


4698 


26 




3b 


I206 


3.29 


5626 


I.34 


558o 


4.64 


4420 


25 




36 


I404 


3.29 


5545 


i-35 


5859 


4.64 


4141 


24 




37 


1602 


3.29 


54 6 5 




6137 


4.64 


3863 


23 




38 


1799 


3- 2 9 


53 8 4 




6415 


4-63 


3585 


22 




39 


1996 


3.28 


53°3 




6693 


4.63 


33°7 


21 




40 
41 


2193 


3.28 


5222 




6971 


4.63 


3029 


20 
19 




9.732390 


3.28 


9.925141 




9.807249 


4.63 


10.192751 




42 


2587 


3.28 


5060 




7527 


4.63 


2473 


18 




43 


2784 


3.28 


4979 




7805 


4-63 


2195 


17 




44 


2980 


3- 2 7 


4897 




8083 


4.63 


1917 


16 




45 


3 X 77 


3- 2 7 


4816 


i-35 


8361 


4-63 


1639 


15 




46 


'3373 


3- 2 7 


4735 


1.36 


8638 


4.62 


1362 


14 




47 


35 6 9 


3- 2 7 


4654 




8916 


4.62 


1084 


13 




48 


3765 


3- 2 7 


457 2 




9193 


4.62 


0807 


12 




49 


3961 


3.26 


4491 




947i 


4.62 


0529 


11 




50 


4157 


3.26 


4409 




9.809748 


4.62 
4.62 


10.190252 


10 
9 




9-734353 


3.26 


9.924328 




9.810025 


10.189975 




52 


4549 


3.26 


4246 




0302 


4.62 


9698 


8 




53 


4744 


3- 2 5 


4164 




0580 


4.62 


9420 


7 




54 


4939 


3- 2 5 


4083 




0857 


4.62 


9143 


6 




55 


5135 


3- 2 5 


4001 




"34 


4.61 


8866 


6 




56 


533° 


3- 2 5 


3919 




1410 


4.61 


8590 


4 




57 


5525 


3- 2 5 


3837 


1.36 


1687 


4.61 


8313 


3 




58 


5719 


3- 2 4 


3755 


i-37 


1964 


4.61 


8036 


2 




59 


59*4 


3- 2 4 


3 6 73 


i-37 


2241 


4.61 


7759 


1 




60 


9.736109 
Cosine. 




9.923591 




9.812517 




10.187483 




M. 




Diff. 1" 


Sine. 


DiffJ" 


Cotang. 


Diff. 1" 


Tang. 




122° 








57° 





74 



33° 


SINES AND TANGENTS. 


146° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 
1-37 


Tang. 
9.812517 


Diff. 1" 


Cotang. 


60 


9.736109 


3- 2 4 


9.923591 


4.61 


10.187483 


1 


6303 


3- 2 4 


3509 




2794 


4.61 


7206 


59 


2 


6498 


3- 2 4 


34 2 7 




3070 


4.61 


6930 


58 


3 


6692 


3- 2 3 


3345 




3347 


4.60 


6653 


57 


4 


6886 


3- 2 3 


3263 




3623 


4.60 


6377 


56 


5 


7080 


3- 2 3 


3181 




3 8 99 


4.60 


6101 


55 


6 


7274 


3- 2 3 


3098 




4175 


4.60 


5825 


54 


7 


7467 


3- 2 3 


3016 




4452 


4.60 


5548 


53 


8 


7661 


3.22 


2933 




4728 


4.60 


5272 


52 


9 


7855 


3.22 


2851 


i-37 


5004 


4.60 


4996 


51 


10 

11 


8048 
9.738241 


3.22 


2768 


1.38 


5279 
9-815555 


4.60 


4721 


50 

49 


3.22 


9.922686 


4-59 


IO.184445 


12 


8434 


3.22 


2603 




5831 


4-59 


4169 


48 


13 


8627 


3.21 


2520 




6107 


4-59 


3893 


47 


14 


8820 


3.21 


2438 




6382 


4-59 


3618 


46 


15 


9013 


3.21 


2-355 




6658 


4-59 


334 2 


45 


16 


9206 


3.21 


2272 




6933 


4-59 


3067 


44 


17 


9398 


3.21 


2189 




7209 


4-59 


2791 


43 


18 


9590 


3.20 


2106 




7484 


4-59 


2516 


42 


19 


9783 


3.20 


2023 




7759 


4-59 


2241 


41 


20 
21 


9-739975 

9.740167 


3.20 


1940 


1.38 
i-39 


8035 


4.58 
4-58 


1965 
10.181690 


40 
39 


3.20 


9.921857 


9.818310 


22 


0359 


3.20 


1774 




8585 


4.58 


1415 


38 


23 


0550 


3-!9 


1691 




8860 


4.58 


1 1 40 


37 


24 


0742 


3-19 


1607 




9*35 


4-58 


0865 


36 


25 


0934 


3-19 


1524 




9410 


4.58 


0590 


35 


26 


1125 


3- J 9 


1 441 




9684 


4.58 


0316 


34 


27 


1316 


3- J 9 


1357 




9.819959 


4.58 


10.180041 


33 


28 


1508 


3.18 


1274 




9.820234 


4.58 


10.179766 


32 


29 


1699 


3.18 


1190 




0508 


4-57 


9492 


31 


30 
31 


1889 
9.742080 


3.18 


1107 


i-39 


0783 
9.821057 


4-57 


9217 
10.178943 


30 
29 


3.18 


9.921023 


4-57 


32 


2271 


3.18 


0939 


1.40 


1332 


4-57 


8668 


28 


33 


2462 


3-17 


0856 




1606 


4-57 


8394 


27 


34 


2652 


3-17 


0772 




1880 


4-57 


8120 


26 


35 


2842 


3-J7 


0688 




2154 


4-57 


7846 


25 


36 


3033 


3-i7 


0604 




2429 


4-57 


757i 


24 


37 


3223 


3- J 7 


0520 




2703 


4-57 


7297 


23 


38 


34i3 


3.16 


0436 




2977 


4.56 


7023 


22 


39 


3602 


3.16 


0352 




3250 


4.56 


6750 


21 


40 
_ 41 


379 2 
9.743982 


3.16 
3.16 


0268 




35 2 4 
9.823798 


4.56 


6476 


20 
19 


9.920184 




4.56 


10.176202 


42 


4171 


3.16 


0099 




4072 


4.56 


5928 


18 


43 


4361 


3-i5 


9.920015 


1.40 


4345 


4.56 


5655 


17 


44 


455o 


3-15 


9.919931 


1.41 


4619 


4.56 


538i 


16 


45 


4739 


3-i5 


9846 




4893 


4.56 


5107 


15 


46 


4928 


3-!5 


9762 




5166 


4.56 


4834 


14 


47 


5117 


3-i5 


9677 




5439 


4-55 


4561 


13 


48 


5306 


3-H 


9593 




5713 


4-55 


4287 


12 


49 


5494 


3-H 


9508 




5986 


4-55 


4014 


11 


50 
51 


5683 
9.745871 


3.14 


9424 




6259 
9.826532 


4-55 


374i 
10.173468 


10 


3-H 


9.919339 


4-55 


52 


6059 


3-H 


9254 




6805 


4-55 


3*95 


8 


53 


6248 


3-*3 


9169 




7078 


4-55 


2922 


7 


54 


6436 


3-!3 


9085 


1.41 


735i 


4-55 


2649 


6 


55 


6624 


3 X 3 


9000 


1.42 


7624 


4-55 


2376 


5 


56 


6812 


3.13 


8915 




7897 


4-54 


2103 


4 


57 


6999 


3-n 


8830 




8170 


4-54 


1830 


3 


58 


7187 


3.12 


8745 




8442 


4-54 


1558 


2 


59 


7374 


3.12 


8659 


1.42 


8715 


4-54 


1285 


1 


60 


9.747562 
Cosine. 


Diff. 1" 


9.918574 




9.828987 
Cotang. 




10.171013 
Tang. 




M. 1 


Sine. 


Diff.l" 


Diff. 1" 


123° 










56° 



75 



34° 




MGARJTHMXC 




145° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.747562 


3.12 


9.918574 


I.42 


9.828987 


4-54 


10.171013 


1 


7749 


3.12 


8489 




9260 


4-54 


0740 


59 


2 


793 6 


3.12 


8404 




9532 


4-54 


0468 


58 


3 


8123 


3-" 


8318 




9.829805 


4-54 


IO.170195 


57 


4 


8310 


3- 11 


8233 




9.830077 


4-54 


IO.169923 


56 


5 


8497 


3-" 


8147 


I.42 


0349 


4-53 


9651 


55 


6 


8683 


3.11 


8062 


I.43 


0621 


4-53 


9379 


54 


7 


8870 


3-" 


7976 




0893 


4-53 


9107 


53 


8 


9056 


3.10 


7891 




1165 


4-53 


8835 


52 


9 


9 2 43 


3.10 


7805 




1437 


4-53 


8563 


51 


10 
11 


9429 


3.10 


7719 
9.917634 




1709 


4-53 


8291 


50 
49 


9.749615 


3.10 




9.831981 


4-53 


10.168019 


12 


9801 


3.10 


7548 




2253 


4-53 


7747 


48 


13 


9.749987 


3-°9 


7462 




2525 


4-53 


7475 


47 


14 


9.750172 


3-°9 


7376 




2796 


4-53 


7204 


46 


15 


0358 


3-°9 


7290 




3068 


4.52 


6932 


45 


16 


0543 


3-°9 


7204 


i-43 


3339 


4.52 


6661 


44 


17 


0729 


3-°9 


7118 


1.44 


3611 


4-5* 


6389 


43 


18 


0914 


3.08 


7032 




3882 


4.52 


6118 


42 


19 


1099 


3.08 


6946 




4154 


4.52 


5846 


41 


20 
21 


1284 


3.08 


6859 
9.916773 




4425 


4.52 


5575 


40 
39 


9.751469 


3.08 




9.834696 


4.52 


10.165304 


22 


1654 


3.08 


6687 




4967 


4.52 


5°33 


38 


23 


1839 


3.08 


6600 




5238 


4.52 


4762 


37 


24 


2023 


3-°7 


6514 




55°9 


4-5 2 


4491 


36 


25 


2208 


3-°7 


6427 




578o 


4-5i 


4220 


35 


26 


2392 


3-°7 


6341 




6051 


4.51 


3949 


34 


27 


2576 


3-°7 


6254 


1.44 


6322 


4.51 


3678 


33 


28 


2760 


3-°7 


6167 


1.45 


6593 


4.51 


3407 


32 


29 


2944 


3.06 


6081 




6864 


4.51 


3136 


31 


30 
31 


3128 
9-7533 12 


3.06 


5994 




7134 


4.51 


2866 


30 
29 


3.06 


9.915907 




9.837405 


4.51 


10.162595 


32 


3495 


3.06 


5820 




7675 


4.51 


2325 


28 


33 


3 6 79 


3.06 


5733 




7946 


4.51 


2054 


27 


34 


3862 


3-°5 


5646 




8216 


4.50 


1784 


26 


35 


4046 


3-°5 


5559 




8487 


4-5° 


1513 


25 


36 


4229 


3-°5 


5472 




8757 


4-5° 


1243 


24 


37 


4412 


3-°5 


5385 




9027 


4.50 


0973 


23 


38 


4595 


3-°5 


5297 




9297 


4.50 


0703 


22 


39 


4778 


3-°4 


5210 


1.45 


9568 


4.50 


0432 


21 


40 
41 


4960 


3-°4 


5123 


1.46 


9.839838 


4.50 


10.160162 


20 
19 


9'755i43 


3-°4 


9-9!5°35 




9.840108 


4.50 


10.159892 


42 


5326 


3-°4 


4948 




0378 


4.50 


9622 


18 


43 


55o8 


3-°4 


4860 




0647 


4.50 


9353 


17 


44 


5690 


3-°4 


4773 




0917 


4.49 


9083 


16 


45 


5872 


3-°3 


4685 




1187 


4.49 


8813 


15 


46 


6054 


3-°3 


4598 




1457 


4.49 


8543 


14 


47 


6236 


3-03 


45i° 




1726 


4.49 


8274 


13 


48 


6418 


3-03 


4422 




1996 


4.49 


8004 


12 


49 


6600 


3-°3 


4334 


1.46 


2266 


4.49 


7734 


11 


50 
51 


6782 


3.02 


4246 
9.914158 


1.47 


2535 


4.49 


7465 


10 

9 


9.756963 


3.02 


9.842805 


4.49 


10.157195 


52 


7144 


3.02 


4070 




3°74 


4.49 


6926 


8 


53 


7326 


3.02 


3982 




3343 


4.49 


6657 


7 


54 


7507 


3.02 


3 8 94 




3612 


4.49 


6388 


6 


55 


7688 


3.01 


- 3806 




3882 


4.48 


6118 


5 


56 


7869 


3.01 


37i8 




4I5 1 


4.48 


5849 


4 


57 


8050 


3.01 


3630 




4420 


4.48 


558o 


3 


58 


8230 


3.01 


354i 




4689 


4.48 


53 11 


2 


59 


8411 


3.01 


3453 


1.47 


4958 


4.48 


5042 


1 


60 


9.758591 




9.913365 




9.845227 
Cotang. 




10.154773 




M. 


Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Diff. \" 


Tang. 


124° 
L 








65° 



76 



35° 


SINES AND TANGENTS. 


144° 


M. 




Sine. 


Diff. 1" 


Cosine. 


Diff.l" 

I.47 


Tang. 


Diff. 1" 


Cotang. 


60 


9-75 8 59i 


3.01 


9-9 J 33 6 5 


9.845227 


4.48 


IO-I54773 


1 


8772 


3.00 


3276 


I.47 


549 6 


4.48 


4504 


59 


2 


8952 


3.00 


3187 


I.48 


5764 


4.48 


4236 


58 


3 


9132 


3.00 


3°99 




6033 


4.48 


39 6 7 


57 


4 


9312 


3.00 


3010 




6302 


4.48 


3698 


56 


5 


9492 


3.00 


2922 




6570 


4-47 


343° 


55 


6 


9672 


2.99 


2833 




6839 


4-47 


3161 


54 


V 


9.759852 


2.99 


2744 




7107 


4-47 


2893 


53 


8 


9.760031 


2.99 


2655 




7376 


4-47 


2624 


52 


9 


02 1 1 


2.99 


2566 




7644 


447 


2356 


51 


10 
11 


0390 


2.99 


2477 
9.912388 


I.48 


79 J 3 
9.848181 


4-47 


2087 


50 
49 


9.760569 


2.98 


4-47 


10.151819 


12 


0748 


2.98 


2299 


I.49 


8449 


4-47 


i55i 


48 


13 


0927 


2.98 


2210 




8717 


4-47 


1283 


47 


14 


1106 


2.98 


2121 




8986 


4-47 


1014 


46 


15 


1285 


2.98 


2031 




9254 


4-47 


0746 


45 


16 


1464 


2.98 


1942 




9522 


4-47 


0478 


44 


17 


1642 


2.97 


1853 




9.849790 


4.46 


10.150210 


43 


18 


1821 


2.97 


1763 




9.850058 


4.46 


10.149942 


42 


19 


1999 


2.97 


1674 




0325 


4.46 


9675 


41 


2D 

21 


2177 
9.762356 


2.97 


1584 




0593 
9.850861 


4.46 


9407 
10.149139 


40 
39 


2.97 


9.911495 




4.46 


22 


2534 


2.96 


1405 


I.49 


1129 


4.46 


8871 


38 


23 


2712 


2.96 


1315 


I.50 


1396 


4.46 


8604 


37 


24 


2889 


2.96 


1226 




1664 


4.46 


8336 


36 


25 


3067 


2.96 


1136 




I93i 


4.46 


8069 


35 


26 


3 2 45 


2.96 


1046 




2199 


4.46 


7801 


34 


27 


3422 


2.96 


0956 




2466 


4.46 


7534 


33 


28 


3600 


2.95 


0866 




2-733 


4-45 


7267 


32 


29 


3777 


2.95 


0776 




3001 


4-45 


6999 


31 


30 


3954 


2.95 


0686 
9.910596 




3268 


4-45 


6732 


30 
29 


31 


9.764131 


2.95 




9- 8 53535 


4-45 


10.146465 


32 


4308 


2.95 


0506 


I.50 


3802 


4-45 


6198 


28 


33 


4485 


2.94 


0415 


1. 51 


4069 


4-45 


593i 


27 


34 


4662 


2.94 


°3 2 5 




433 6 


4-45 


5664 


26 


35 


4838 


2.94 


0235 




4603 


4-45 


5397 


25 


36 


5°i5 


2.94 


0144 




4870 


4-45 


5^° 


24 


37 


5i9i 


2.94 


9.910054 




5137 


4-45 


4863 


23 


38 


53 6 7 


2.94 


9.909963 




5404 


4-45 


4596 


22 


39 


5544 


*-93 


9 8 73 




5 6 7i 


4.44 


4329 


21 


40 
41 


5720 


2.93 


9782 
9.909691 




5938 
9.856204 


4.44 


4062 
10.143796 


20 
19 


9.765896 


2 -93 




4.44 


42 


6072 


2.93 


9601 




6471 


4.44 


35 2 9 


18 


43 


6247 


^•93 


9510 




6737 


4.44 


3263 


17 


44 


6423 


*-93 


9419 


i-5i 


7004 


4.44 


2996 


16 


45 


6598 


2.92 


9328 


1.52 


7270 


4.44 


2730 


15 


46 


6774 


2.92 


9237 




7537 


4.44 


2463 


14 


47 


6949 


2.92 


9146 




7803 


4.44 


2197 


13 


48 


7124 


2.92 


9°55 




8069 


4.44 


!93i 


12 


49 


7300 


2.92 


8964 




8336 


4.44 


1664 


11 


50 


7475 


2.91 


8873 




8602 


4-43 


1398 


10 


51 


9.767649 


2.91 


9.908781 




9.858868 


4-43 


10.141132 


9 


52 


7824 


2.91 


8690 




9 J 34 


4-43 


0866 


8 


53 


7999 


2.91 


8599 




9400 


4-43 


0600 


7 


54 


8173 


2.91 


8507 


1.52 


9666 


4-43 


°334 


6 


55 


8348 


2.90 


8416 


i-53 


9.859932 


4-43 


10.140068 


5 


56 


8522 


2.90 


8324 




9.860198 


4-43 


10.139802 


4 


57 


8697 


2.90 


8233 




0464 


4-43 


9536 


3 


58 


8871 


2.90 


8141 




0730 


4-43 


9270 


2 


59 


9045 


2.90 


8049 


i-53 


0995 


4-43 


9005 


1 


60 


9.769219 
Cosine. 




9.907958 

Sine. 


Diff.l" 


9.861261 
Cotang. 




10.138739 



M. 


Diff. 1" 


Diff. 1" 


Tang. 


125° 










54° 



/ 1 



36° 




XiOaASlZTHMXC 




143° 


M. 



Sine. 


Diff. 1" 
2.90 


Cosine. 


Diff.l" 
1-53 


Tang. 


Diff. 1" 


Cotang. * 


60 


9.769219 


9.907958 


9.861261 


4-43 


10.138739 


i 


9393 


2.89 


7866 




1527 


4-43 


8473 


59 


2 


9566 


2.89 


7774 




1792 


4.42 


8208 


58 


3 


974° 


2.89 


7682 




2058 


4.42 


7942 


57 


4 


9.769913 


2.89 


759° 




2323 


4.42 


7677 


56 


5 


9.770087 


2.89 


7498 




2589 


4.42 


741 1 


55 


6 


0260 


2.88 


7406 


i-53 


2854 


4.42 


7146 


54 


7 


0433 


2.88 


73*4 


1.54 


3"9 


4.42 


6881 


53 


8 


0606 


2.88 


7222 




3385 


4.42 


6615 


52 


9 


0779 


2.88 


7129 




3650 


4.42 


6350 


51 


10 
11 


0952 


2.88 


7°37 




39 J 5 


4.42 


6085 


50 
49 


9.771125 


2.88 


9.906945 




9.864180 


4.42 


10.135820 


12 


1298 


2.88 


6852 




4445 


4.42 


5555 


48 


13 


1470 


2.87 


6760 




4710 


4.42 


5290 


47 


14 


1643 


2.87 


6667 




4975 


4.41 


5° 2 5 


46 


15 


1815 


2.87 


6575 




5240 


4.41 


4760 


45 


16 


1987 


2.87 


6482 


1.54 


55°5 


4.41 


4495 


44 


17 


2159 


2.87 


6389 


i-55 


577o 


4.41 


4230 


43 


18 


2331 


2.86 


6296 




6035 


4.41 


39 6 5 


42 


19 


2503 


2.86 


6204 




6300 


4.41 


3700 


41 


20 
21 


2675 


2.86 


6111 




6564 


4.41 


3436 


40 
39 


9.772847 


2.86 


9.906018 




9.866829 


4.41 


10.133171 


22 


3018 


2.86 


5925 




7094 


4.41 


2906 


38 


23 


3190 


2.86 


5832 




7358 


4.41 


2642 


37 


24 


33 61 


2.85 


5739 




7623 


4.41 


2377 


36 


25 


3533 


2.85 


5 6 45 




7887 


4.41 


2113 


35 


26 


3704 


2.85 


555 2 




8152 


4.40 


1848 


34 


27 


3^75 


2.85 


5459 


i-55 


8416 


4.40 


1584 


33 


28 


4046 


2.85 


5366 


1.56 


8680 


4.40 


1320 


32 


29 


4217 


2.85 


5272 




8945 


4.40 


i°55 


31 


30 
31 


43^ 
9-77455 8 


2.84 


5179 




9209 


4.40 


0791 


30 

29 


2.84 


9.905085 




9473 


4.40 


0527 


32 


4729 


2.84 


4992 




9.869737 


4.40 


10.130263 


28 


33 


4899 


2.84 


4898 




9.870001 


4.40 


10.129999 


27 


34 


5070 


2.84 


4804 




0265 


4.40 


9735 


26 


35 


5240 


2.84 


471 1 




0529 


4.40 


947i 


25 


36 


54io 


2.83 


4617 




0793 


• 4.40 


9207 


24 


37 


558o 


2.83 


45*3 


1.56 


1057 


4.40 


8943 


23 


38 


575° 


2.83 


4429 


i-57 


1321 


4.40 


8679 


22 


39 


5920 


2.83 


4335 




1585 


4.40 


8415 


21 


40 
41 


6090 


2.83 


4241 




1849 


4-39 


8151 


20 
19 


9.776259 


2.83 


9.904147 




9.872112 


4-39 


10.127888 


42 


6429 


2.82 


4053 




2376 


4-39 


7624 


18 


43 


6598 


2.82 


3959 




2640 


4-39 


7360 


17 


44 


6768 


2.82 


3864 




2903 


4-39 


7097 


16 


45 


6937 


2.82 


377o 




3167 


4-39 


6833 


15 


46 


7106 


2.82 


3676 




343° 


4-39 


6570 


14 


47 


7275. 


2.81 


358i 




3 6 94 


4-39 


6306 


13 


48 


7444 


2.81 


3487 


i-57 


. 3957 


4-39 


6043 


12 


49 


7613 


2.81 


339 2 


1.58 


4220 


4-39 


578o 


11 


50 
51 


7781 


2.81 


3298 




4484 


4-39 


55I 6 


10 

9 


9.777950 


2.81 


9.903203 




9.874747 


4-39 


10.125253 


52 


8119 


2.81 


3108 




5010 


4-39 


4990 


8 


53 


8287 


2.80 


3014 




5 2 73 


4.38 


4727 


7 


54 


8455 


2.80 


2919 




553 6 


4-38 


4464 


6 


55 


8624 


2.80 


2824 




5800 


4-38 


4200 


5 


56 


8792 


2.80 


2729 




6063 


4.38 


3937 


4 


57 


8960 


2.80 


2634 




6326 


4.38 


3 6 74 


3 


58 


9128 


2.80 


2539 


1.58 


6589 


4-38 


34" 


2 


59 


9295 


2.79 


2444 


1.59 


6851 


4.38 


3*49 


1 


60 


9-7794 6 3 




9.902349 


Diff.l" 


9.877114 




10.122886 




M. 


Cosine. 


Diff. 1" 


Sine. 


Cotang. 


Diff. 1" 


Tang. 


126° 








53° 



78 





37° 


SXtfES AND TANGENTS. 


142° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 
IO.122886 


60 


9.779463 


2.79 


9.902349 


I.59 


9.877114 


4.38 


J 


9631 


2.79 


2253 




7377 


4.38 


2623 


59 


2 


9798 


2.79 


2158 




7640 


4-38 


2360 


58 


8 


9.779966 


2.79 


2063 




7903 


4-38 


2097 


57 


4 


9.780133 


2.79 


1967 




8165 


4-38 


1835 


56 


b 


0300 


2.78 


1872 




8428 


4.38 


1572 


55 


6 


0467 


2.78 


1776 




8691 


4.38 


1309 


54 


7 


0634 


2.78 


1681 




8953 


4-37 


1047 


53 ; 


8 


0801 


2.78 


1585 




9216 


4-37 


0784 


52 


y 


0968 


2.78 


I490 


I.59 


9478 


4-37 


0522 


51 ! 


10 

n 


"34 


2.78 


1394 


I.60 


9.879741 
9.880003 


4-37 


10.120259 


50 

49 


9.781301 


2.77 


9.901298 




4-37 


IO.II9997 


12 


1468 


2.77 


1202 




0265 


4-37 


9735 


48 


13 


1634 


2.77 


1 106 




0528 


4-37 


9472 


47 


14 


1800 


2.77 


IOIO 




0790 


4-37 


9210 


46 


15 


1966 


2.77 


0914 




1052 


4-37 


8948 


45 


16 


2132 


2.77 


0818 




J 3 J 4 


4-37 


8686 


44 


17 


2298 


2.76 


0722 




1576 


4-37 


8424 


43 


18 


2464 


2.76 


0626 




1839 


4-37 


8161 


42 


19 


2630 


2.76 


0529 


I.60 


2101 


4-37 


7899 


41 


20 
21 


2796 


2.76 


0433 
9.900337 


I.61 


2363 


4-3 6 


7637 


40 
39 


9.782961 


2.76 




9.882625 


4.36 


10.117375 


22 


3127 


2.76 


0240 




2887 


4.36 


7113 


38 


23 


3292 


2.75 


0144 




3148 


4.36 


6852 


37 


24 


3458 


2.75 


9.900047 




3410 


4.36 


6590 


36 


25 


3623 


2.75 


9.899951 




3672 


4.36 


6328 


35 


26 


3788 


2.75 


9854 




3934 


4-3 6 


6066 


34 


27 


3953 


2.75 


9757 




4196 


4.36 


5804 


33 


28 


4118 


2.74 


9660 




4457 


4.36 


5543 


32 


29 


4282 


2.74 


9564 


I.6l 


4719 


4-3 6 


5281 


31 


30 
31 


4447 
9.784612 


2.74 


9467 


I.62 


4980 
9.885242 


4-3 6 


5020 


30 
29 


2.74 


9.899370 


4-3 6 


10.114758 


82 


4776 


2.74 


9273 




5503 


4-3 6 


4497 


28 


88 


4941 


2.74 


9176 




5765 


4-3 6 


4235 


27 


84 


5i°5 


2.74 


9078 




6026 


4-3 6 


3974 


26 


35 


5269 


2.73 


8981 




6288 


4.36 


3712 


25 


36 


5433 


2.73 


8884 




6549 


4-35 


3451 


24 


37 


5597 


2-73 


8787 




6810 


4-35 


3190 


23 


38 


5761 


2 -73 


8689 




7072 


4-35 


2928 


22 


39 


59 2 5 


2-73 


8592 


I.62 


7333 


4-35 


2667 


21 


40 
41 


6089 


2.73 


8494 


I.63 


7594 


4-35 


2406 


20 
19 


9.786252 


2.72 


9.898397 




9.887855 


4-35 


10.112145 


42 


6416 


2.72 


8299 




8116 


4-35 


1884 


18 


43 


6 579 


2.72 


8202 




8377 


4-35 


1623 


17 


44 


6742 


2.72 


8104 




8639 


4-35 


1361 


16 


45 


6906 


2.72 


8006 




8900 


4-35 


1 1 00 


15 


46 


7069 


2.72 


7908 




9160 


4-35 


0840 


14 


47 


7232 


2.71 


7810 




9421 


4-35 


0579 


13 


48 


7395 


2.71 


7712 




9682 


4-35 


0318 


12 


49 


7557 


2.71 


7614 




9.889943 


4-35 


10.110057 


11 


50 
51 


7720 


2.71 


7516 
9.897418 


I.63 


9.890204 


4-34 


10.109796 


10 
~~9~ 


9.787883 


2.71 


I.64 


0465 


4-34 


9535 


52 


8045 


2.71 


7320 




0725 


4-34 


9275 


8 


53 


8208 


2.71 


7222 




0986 


4-34 


9014 


7 


54 


8370 


2.70 


7123 




1247 


4-34 


8753 


6 


55 


8532 


2.70 


7025 




1507 


4-34 


8493 


5 


56 


8694 


2.70 


6926 




1768 


4-34 


8232 


4 


57 


8856 


2.70 


6828 




2028 


4-34 


7972 


3 


58 


9018 


2.70 


6729 




2289 


4-34 


7711 


2 


59 


9180 


2.70 


6631 


I.64 


2549 


4-34 


745i 


1 


60 


9.789342 
Cosine. 




9-89653 2 
Sine. 




9.892810 




10.107190 



M. 


Diff. 1" 


Diff.l" 


Cotang. 


Diff. 1" 


Tang. 


127° 








52° 



79 



38° 




LOGARITHMIC 




141° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.789342 


2.69 


9.896532 


I.64 


9.892810 


4-34 


10.107190 


1 


9504 


2.69 


6 433 


I.65 


3070 


4-34 


6930 


50 


a 


9665 


2.69 


6 335 




333 1 


4-34 


6669 


58 


3 


9827 


2.69 


6236 




359 1 


4-34 


6409 


57 


4 


9.789988 


2.69 


6137 




3851 


4-34 


6149 


56 


5 


9.790149 


2.69 


6038 




4111 


4-34 


5889 


55 


6 


0310 


2.68 


5939 




437i 


4-34 


5629 


54 


7 


0471 


2.68 


5840 




4632 


4-33 


5368 


53 


8 


0632 


2.68 


574i 




4892 


4-33 


5108 


52 


y 


0793 


2.68 


5 6 4i 




5152 


4-33 


4848 


51 


10 

n 


0954 


2.68 


5542 


I.65 
1.66 


5412 


4-33 


4588 


50 
40 


9.791115 


2.68 


9- 8 95443 


9.895672 


4-33 


IO.I04328 


12 


1275 


2.67 


5343 




5932 


4-33 


4068 


48 


13 


1436 


2.67 


5244 




6192 


4-33 


3808 


47 


14 


1596 


2.67 


5H5 




6452 


4-33 


3548 


46 


15 


1757 


2.67 


5°45 




6712 


4-33 


32.88 


45 


16 


1917 


2.67 


4945 




6971 


4-33 


3029 


44 


17 


2077 


2.67 


4846 




7231 


4-33 


2769 


43 


18 


2237 


2.66 


4746 




749 1 


4-33 


2509 


42 


iy 


2 397 


2.66 


4646 




775 1 


4-33 


2249 


41 


20 
21 


2 557 
9.792716 


2.66 


4546 


1.66 


8010 


4-33 


1990 


40 

30 


2.66 


9.894446 


1.67 


9.898270 


4-33 


10.101730 


22 


2876 


2.66 


4346 




8530 


4-33 


1470 


38 


23 


3°35 


2.66 


4246 




8789 


4-3 2 


I2II 


37 


24 


3195 


2.66 


4146 




9049 


4-3 2 


O951 


36 


25 


3354 


2.65 


4046 




93°8 


4-3 2 


0692 


35 


26 


35H 


2.65 


3946 




9568 


4.32 


O432 


34 


27 


3 6 73 


2.65 


3846 




9.899827 


4-3 2 


IO.IOOI73 


33 


28 


3832 


2.65 


3745 




9.900086 


4.32 


IO.O99914 


32 


2y 


399i 


2.65 


3 6 45 




0346 


4.32 


9654 


31 


30 
31 


4150 


2.64 


3544 


1.67 


0605 


4.32 


9395 


30 
20 


9.794308 


2.64 


9.893444 


1.68 


9.900864 


4.32 


10.099136 


32 


4467 


2.64 


3343 




1 1 24 


4-3 2 


8876 


28 


33 


4626 


2.64 


3 2 43 




1383 


4-3 2 


8617 


27 


34 


4784 


2.64 


3 J 4 2 




1642 


4-3 2 


8358 


26 


35 


4942 


2.64 


3041 




1901 


4-3 2 


8099 


25 


36 


5101 


2.64 


2940 




2160 


4-3 2 


7840 


24 


37 


5 2 59 


2.63 


2839 




2419 


4-3 2 


7581 


23 


38 


54 x 7 


2.63 


2739 




2679 


4-3 2 


7321 


22 


3y 


5575 


2.63 


2638 




2938 


4-3 2 


7062 


21 


40 
41 


5733 


2.63 


2536 


1.68 


3 J 97 


4.31 


6803 
10.096545 


20 
19 


9.795891 


2.63 


9.892435 


1.69 


9-9°3455 


4-3 1 


42 


6049 


2.63 


2334 




37H 


4-3i 


6286 


18 


43 


6206 


2.63 


2233 




3973 


4-3 1 


6027 


17 


44 


6364 


2.62 


2132 




4232 


4-3i 


5768 


16 


45 


6521 


2.62 


2030 




449 1 


4-3i 


5509 


15 


46 


6679 


2.62 


1929 




475o 


4-3 1 


5 2 5° 


14 


47 


6836 


2.62 


1827 




5008 


4-3 J 


4992 


13 


48 


6993 


2.62 


1726 




5267 


4.31 


4733 


12 


4y 


7150 


2.62 


1624 


1.69 


55 26 


4-3 1 


4474 


11 


50 
51 


7307 


2.61 


1523 


1.70 


5784 


4-3i 


4216 
10.093957 


10 



9.797464 


2.61 


9.89 1 42 1 




9.906043 


4-3 1 


52 


7621 


2.61 


1319 




6302 


4-3 1 


3698 


8 


53 


7777 


2.61 


1217 




6560 


4-3 x 


344° 


7 


54 


7934 


2.61 


1115 




6819 


4.31 


3181 


6 


55 


8091 


2.61 


1013 




7077 


4.31 


2923 


5 


56 


8247 


2.61 


0911 




733 6 


4-3i 


2664 


4 


57 


8403 


2.60 


0809 




7594 


4.31 


2406 


3 


58 


8560 


2.60 


0707 




7852 


4.31 


2148 


2 j 


50 


8716 


2.60 


0605 


1.70 


8m 


4-3° 


1889 


1 


60 


9.798872 




9.890503 




9.908369 




10.091631 




M. | 


Cosine. 


Diff. 1" 


Sine. 


Diff.]" 


Cotang. 


Diff. 1" 


Tang. 


128° 








51° 



80 






I 39° 


SISTES AWTD TAEJG-ENTS. 


140° 


j M. 




Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.798872 


2.60 


9.890503 


I.70 


9.908369 


4.30 


10.091631 


i 


9028 


2.60 


0400 


I.7I 


8628 


4.30 


1372 


59 


2 


9184 


2.60 


0298 




8886 


4.30 


1114 


58 


3 


9339 


2.59 


0195 




9144 


4.30 


0856 


57 


4 


9495 


2.59 


9.890093 




9402 


4-3° 


0598 


56 


5 


9651 


2.59 


9.889990 




9660 


4-3° 


0340 


55 


6 


9806 


2.59 


9888 




9.909918 


4.30 


10.090082 


54 


7 


9.799962 


2.59 


9785 




9.910177 


4.30 


IO.089823 


53 


8 


9.800117 


2.59 


9682 




0435 


4.30 


9565 


52 


y 


0272 


2.58 


9579 




0693 


4.30 


9307 


51 


10 

n 


0427 


2.58 


9477 


I.71 
I.72 


0951 
9.911209 


4-3° 


9049 
10.088791 


50 
49 


9.800582 


2.58 


9.889374 


4-3° 


12 


0737 


2.58 


9271 




1467 


4.30 


8533 


48 


18 


0892 


2.58 


9168 




1724 


4.30 


8276 


47 


14 


1047 


2.58 


9064 




1982 


4-3° 


8018 


46 


15 


1201 


2.58 


8961 




2240 


4-3° 


7760 


45 


16 


I35 6 


2.57 


8858 




2498 


4.30 


7502 


44 


17 


1511 


2.57 


8755 




2756 


4.30 


7244 
6986 


43 


18 


1665 


2.57 


8651 




3014 


4.29 


42 


19 


1819 


2.57 


8548 


I.72 


3271 


4.29 


6729 


41 


20 
21 


1973 
9.802128 


2.57 


8444 


I.73 


352-9 


4.29 
4.29 


6471 
IO.086213 


40 
39 


2.57 


9.888341 




9.913787 


22 


2282 


2.56 


8237 




4044 


4.29 


595 6 


38 


23 


2436 


2.56 


8134 




4302 


4.29 


5698 


37 


24 


2589 


2.56 


8030 




4560 


4.29 


5440 


36 


25 


2743 


2.56 


7926 




4817 


4.29 


5183 


35 


26 


2897 


2.56 


7822 




5075 


4.29 


4925 


34 


27 


3050 


2.56 


7718 




5332 


4.29 


4668 


33 


28 


3204 


2.56 


7614 




559° 


4.29 


4410 


32 


29 


3357 


2.55 


7510 


1-73 


5847 


4.29 


4153 


31 


30 
31 


3511 


2.55 


7406 


i-74 


6104 


4.29 
4.29 


3896 
10.083638 


30 
29 


9.803664 


2-55 


9.887302 


9.916362 


32 


3817 


2.55 


7198 




6619 


4.29 


338i 


28 


33 


397° 


^•55 


7°93 




6877 


4.29 


3123 


27 


34 


4123 


2.55 


6989 




7134 


4.29 


2866 


26 


35 


4276 


2.54 


6885 




739i 


4.29 


2609 


25 


36 


4428 


2.54 


6780 




7648 


4.29 


2352 


24 


37 


4581 


2.54 


6676 




7905 


4.29 


2095 


23 


38 


4734 


2.54 


6571 




8163 


4.28 


1837 


22 


39 


4886 


2-54 


6466 


1.74 


8420 


4.28 


1580 


21 


40 
41 


5°39 


2.54 
2.54 


6362 
9.886257 


'•75 


8677 
9.918934 


4.28 


1323 


20 
~19~ 


9.805191 


4.28 


10.081066 


42 


5343 


2-53 


6152 




9191 


4.28 


0809 


18 


43 


5495 


2-53 


6047 




9448 


4.28 


0552 


17 


44 


5 6 47 


^•53 


5942 




9705 


4.28 


0295 


16 


45 


5799 


2-53 


5837 




9.919962 


4.28 


10.080038 


15 


46 


595i 


2-53 


5732 




9.920219 


4.28 


10.079781 


14 


47 


6103 


^•53 


5627 




0476 


4.28 


9524 


13 


48 


6254 


2-53 


5522 




0733 


4.28 


9267 


12 


49 


6406 


2.52 


54i 6 


1-75 


0990 


4.28 


9010 


11 


50 
51 


6 557 
9.806709 


2.52 
2.52 


53 11 


1.76 


1247 
9.921503 


4.28 


8753 
10.078497 


10 
9 


9.885205 


4.28 


52 


6860 


2.52 


5100 




1760 


4.28 


8240 


8 


53 


7011 


2.52 


4994 




2017 


4.28 


7983 


7 


54 


7163 


2.52 


4889 




2274 


4.28 


7726 


6 


55 


73H 


2.52 


4783 




2530 


4.28 


7470 


5 


56 


7465 


2.51 


4677 




2787 


4.28 


7213 


4 


! 57 


7615 


2.51 


4572 


1.76 


3°44 


4.28 


6956 


3 


58 


7766 


2.51 


4466 


1.77 


3300 


4.28 


6700 


2 


59 


7917 


2.51 


4360 


1.77 


3557 


4.27 


6443 


1 I 


60 

1 


9.808067 
Cosine. 


Diff. 1" 


9.884254 


Diff.l" 


9.923813 
Cotang. 




10.076187 
Tang. 


\ 
M. 1 


Sine. 


Diff. 1" 


129° 








50° 



81 



40° 




£Oa.ARXTHIftIC 




139° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.808067 


2.51 


9.884254 


I.77 


9.923813 


4.28 


IO.076187 


1 


8218 


2.51 


4148 




4070 


4.27 


593° 


59 


2 


8368 


2.51 


4042 




4327 


4.27 


5 6 73 


58 


3 


8519 


2.50 


393 6 




4583 


4.27 


5417 


57 


4 


8669 


2.50 


3829 




4840 


4.27 


5160 


56 


5 


8819 


2.50 


3723 




5096 


4.27 


4904 


55 


6 


8969 


2.50 


3617 




535* 


4.27 


4648 


54 


7 


9119 


2.50 


35 IG 




5609 


4.27 


4391 


53 


8 


9269 


2.50 


34°4 


I.77 


5865 


4.27 


4135 


52 


9 


9419 


2.49 


3 2 97 


I.78 


6122 


4.27 


3878 


51 


10 
11 


9569 


2.49 


3I9 1 





6378 


4.27 


3622 


50 
49 


9718 


2.49 


9.883084 


9.926634 


4.27 


10.073366 


12 


9.809868 


2.49 


2977 




6890 


4.27 


3110 


48 


13 


9.810017 


2.49 


2871 




7147 


4.27 


2853 


47 


14 


0167 


2.49 


2764 




7403 


4.27 


2597 


46 


15 


0316 


2.48 


2657 




7659 


4.27 


2341 


45 


16 


0465 


2.48 


2550 




79*5 


4.27 


2085 


44 


17 


0614 


2.48 


2443 


I.78 


8171 


4.27 


1829 


43 


18 


0763 


2.48 


233 6 


I.79 


8427 


4.27 


1573 


42 


19 


0912 


2.48 


2229 




8683 


4.27 


1317 


41 


20 
21 


1061 


2.48 


2121 




8940 


4.27 


1060 


40 
39 


9.811210 


2.48 


9.882014 




9.929196 


4.27 


10.070804 


22 


1358 


2.48 


1907 




9452 


4.27 


0548 


38 


23 


1507 


2.47 


1799 




9708 


4.27 


0292 


37 


24 


1655 


2.47 


1692 




9.929964 


4.27 


10.070036 


36 


2b 


1804 


2.47 


1584 




9.930220 


4.26 


10.069780 


35 


26 


1952 


2.47 


1477 




o475 


4.26 


95 2 5 


34 


27 


2IOO 


2.47 


1369 


I.79 


0731 


4.26 


9269 


33 


28 


2248 


2.47 


1261 


I.80 


0987 


4.26 


9013 


32 


29. 


2396 


2.46 


"53 




1243 


4.26 


8757 


31 


30 
31 


2544 
9.812692 


2.46 


1046 




1499 


4.26 


8501 


30 
29 


2.46 


9.880938 




9-93 I 755 


4.26 


10.068245 


32 


2840 


2.46 


0830 




2010 


4.26 


7990 


28 


33 


2988 


2.46 


0722 




2266 


4.26 


7734 


27 


34 


3 J 35 


2.46 


0613 




2522 


4.26 


7478 


26 


35 


3283 


2.46 


0505 




2778 


4.26 


7222 


25 


36 


343° 


2.46 


0397 


I.80 


3°33 


4.26 


6967 


24 


37 


3578 


2.45 


0289 


I.81 


3289 


4.26 


6711 


23 


38 


3725 


2.45 


0180 




3545 


4.26 


6 455 


22 


39 


3872 


2.45 


9.880072 




3800 


4.26 


6200 


21 


40 
41 


4019 


2.45 


9.879963 




4056 


4.26 


5944 


20 
19 


9.814166 


2.45 


9855 


9.934311 


4.26 


10.065689 


42 


43 J 3 


2.45 


9746 




45 6 7 


4.26 


5433 


18 


43 


4460 


2.44 


9 6 37 




4823 


4.26 


5 J 77 


17 


44 


4607 


2.44 


9529 




5078 


4.26 


4922 


16 


45 


4753 


2.44 


9420 




5333 


4.26 


4667 


15 


46 


4900 


2.44 


93" 


1. 8l 


5589 


4.26 


441 1 


14 


47 


5046 


2.44 


9202 


I.82 


5844 


4.26 


4156 


13 1 


48 


5*93 


2.44 


9°93 




6100 


4.26 


3900 


12 1 


49 


5339 


2.44 


8984 




6 355 


4.26 


3645 


11 ! 


50 
51 


54^5 


2-43 


8875 




6610 


4.26 


339° 


10 

9 


9.815631 


2-43 


9.878766 


9.936866 


4.25 


10.06-3134 


52 


5778 


2.43 


8656 




7121 


4.25 


2879 


8 


53 


59 2 4 


2.43 


8547 




7376 


4.25 


2624 


7 


54 


6069 


2.43 


. 8438 




7632 


4.25 


2368 


6 


55 


6215 


^•43 


8328 


I.82 


7887 


4.25 


2113 


5 


56 


6361 


2-43 


8219 


I.83 


8142 


4.25 


1858 


4 


57 


6507 


2.42 


8109 




8398 


4.25 


1602 


3 


58 


6652 


2.42 


7999 




8653 


4.25 


1347 


2 


59 


6798 


2.42 


7890 


I.83 


8908 


4.25 


1092 


1 


60 


9.816943 




9.877780 




9.939163 




10.060837 



M. 


Cosine. 


Diff. 1" 


Sine. 


Diff.l" 


Cotang. 


Diff. 1" 


Tang. 


130° 








49° 



82 



41° 


SINES AND TANGENTS. 


138° 


M. 

~~ 


Sine. 
9.816943 


Diff. 1" 


Cosine. 


Diff.l" 
I.83 


Tang. 


Diff. 1" 


Cotang. 


60 


2.42 


9.877780 


9.939163 


4.25 


IO.060837 


1 


7088 


2.42 


7670 




9418 


4.25 


0582 


59 


2 


7233 


2.42 


7560 




9 6 73 


4-25. 


0327 


58 


3 


7379 


2.42 


745° 




9.939928 


4.25 


10.060072 


57 


4 


7524 


2.42 


734° 


I.83 


9.940183 


4.25 


10.059817 


56 


5 


7668 


2.41 


7230 


I.84 


0438 


4-25 


9562 


55 


6 


7813 


2.41 


7120 




0694 


4.25 


9306 


54 


7 


7958 


2.41 


7010 




0949 


4.25 


9051 


53 


8 


8103 


2.41 


6899 




1204 


4-25 


8 79 6 


52 


9 


8247 


2.41 


6789 




1458 


4.25 


8542 


51 


10 
11 


8392 


2.41 


6678 




1714 


4-25 


8286 


50 
49 


9.818536 


2.40 


9.876568 


9.941968 


4.25 


10.058032 


12 


8681 


2.40 


6457 




2223 


4.25 


7777 


48 


13 


8825 


2.40 


6347 


I.84 


2478 


4.25 


7522 


47 


14 


8969 


2.40 


6236 


I.85 


2733 


4.25 


7267 


46 


16 


9113 


2.40 


6125 




2988 


4.25 


7012 


45 


16 


9257 


2.40 


6014 




3243 


4.25 


6757 


44 


17 


9401 


2.40 


59°4 




349 8 


4.25 


6502 


43 


18 


9545 


2.40 


5793 




3752 


4.25 


6248 


42 


19 


9689 


2-39 


5682 




4007 


4.25 


5993 


41 


20 
21 


9832 


2.39 


557i 




4262 


4.25 


5738 


40 
39 


9.819976 


2-39 


9.875459 




9.944517 


4.25 


10.055483 


22 


9.820120 


2.39 


5348 




477 1 


4.24 


5229 


38 


23 


0263 


2-39 


5237 


I.85 


5026 


4.24 


4974 


37 


24 


0406 


2-39 


5126 


1.86 


5281 


4.24 


4719 


36 


25 


0550 


2.38 


5014 




5535 


4.24 


4465 


35 


26 


0693 


2.38 


4903 




579° 


4.24 


4210 


34 


27 


0836 


2.38 


479i 




6045 


4.24 


3955 


33 


28 


0979 


2.38 


4680 




6299 


4.24 


3701 


32 


29 


1 1 22 


2.38 


4568 




6554 


4.24 


3446 


31 


! 30 
31 


1265 


2.38 


445 6 




6808 


4.24 


3192 


30 
29 


9.821407 


2.38 


9- 8 74344 


1.86 


9.947063 


4.24 


10.052937 


32 


1550 


2.38 


4232 


1.87 


7318 


4.24 


2682 


28 


33 


1693 


2.37 


4121 




7572 


4.24 


2428 


27 


34 


1835 


2.37 


4009 




7826 


4.24 


2174 


26 


35 


1977 


2-37 


3896 




8081 


4.24 


1919 


25 


36 


2120 


2-37 


3784 




8336 


4.24 


1664 


24 


37 


2262 


2-37 


3672 




8590 


4.24 


1410 


23 


38 


2404 


2-37 


3560 




8844 


4.24 


1 156 


22 


39 


2546 


2.37 


344 8 




9099 


4.24 


0901 


21 


40 
41 


2688 


2.36 


3335 




9353 


4.24 


0647 


20 
19 


9.822830 


2.36 


9.873223 


1.87 


9607 


4.24 


°393 


42 


2972 


2.36 


3110 


1.88 


9.949862 


4.24 


10.050138 


18 


43 


3"4 


2.36 


2998 




9.950116 


4.24 


10.049884 


17 


44 


32-55 


2.36 


2885 




0370 


4.24 


9630 


16 


45 


3397 


2.36 


2772 




0625 


4.24 


9375 


15 


46 


3539 


2.36 


2659 




0879 


4.24 


9121 


14 


47 


3680 


2-35 


2547 




"33 


4.24 


8867 


13 


48 


3821 


2-35 


2434 




1388 


4.24 


8612 


12 


49 


39 6 3 


2-35 


2321 




1642 


4.24 


8358 


11 


50 
51 


4104 


2 -35 
2-35 


2208 
9.872095 


1.88 


1896 


4.24 


8104 


10 
9 


9.824245 


1.89 


9.952150 


4.24 


10.047850 


52 


4386 


2-35 


1981 




2405 


4.24 


7595 


8 


53 


45 2 7 


2.35 


1868 




2659 


4.24 


734i 


7 


54 


4668 


2.34 


1755 




2913 


4.24 


7087 


6 


55 


4808 


2-34 


1641 




3167 


4-23 


6833 


5 


56 


4949 


2-34 


1528 




3421 


4.23 


6579 


4 


57 


5090 


2-34 


1414 




3 6 75 


4.23 


6345 


3 


58 


5230 


2-34 


1301 




3929 


4.23 


6071 


2 


59 


537i 


2-34 


1187 


1.89 


4183 


4.23 


5817 


1 


60 


9.825511 




9.871073 
Sine. 


Diff.l" 


9-954437 




10.045563 

Tang. 



M. 


Cosine. 


Diff. 1" 


Cotang. 


Diff. 1" 


131° 

... 








48° 



83 



| 42° 




IiOaAXUTHMXC 




137° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff. 1" 


Tang. 


Diff. 1" 


Cotang. 


60 


9.825511 


2.34 


9.871073 


I.90 


9-954437 


4.23 


IO.045563 


1 


5 6 5l 


2 -33 


0960 




4691 


4.23 


53°9 


59 


2 


5791 


*-33 


0846 




4945 


4.23 


5°55 


58 


3 


5931 


2 -33 


0732 




5200 


4.23 


4800 


57 


4 


6071 


2 -33 


0618 




5454 


4.23 


4546 


56 


5 


6211 


2 -33 


0504 




57°7 


4.23 


4293 


55 


6 


6351 


2 -33 


0390 




5961 


4.23 


4°39 


54 


V 


6491 


2 -33 


0276 




6215 


4-2 3 


3785 


53 


8 


6631 


2 -33 


0161 


I.90 


6469 


4.23 


353i 


52 


y 


6770 


2.32 


9.870047 


I.91 


6723 


4- 2 3 


3 2 77 


51 


10 

n 


6910 


2.32 


9.869933 




6977 


4.23 


3023 


50 
49 • 


9.827049 


2.32 


9818 


9.957231 


4-23 


10.042769 


12 


7189 


2.32 


9704 




7485 


4-^3 


2515 


48 


13 


7328 


2.32 


9589 




7739 


4.23 


2261 


47 


14 


7467 


2.32 


9474 




7993 


4.23 


2007 


46 


15 


7606 


2.32 


9360 




8246 


4-23 


1754 


45 


16 


7745 


2.32 


9245 




8500 


4-23 


1500 


44 


17 


7884 


2.31 


9130 


I.91 


8754 


4-*3 


1246 


43 


18 


8023 


2.31 


9015 


I.92 


9008 


4.23 


0992 


42 


19 


8162 


2.31 


8900 




9262 


4.23 


0738 


41 


20 
~21 


8301 


2.31 


8785 




9516 


4.23 


0484 


40 
39 


9.828439 


2.31 


9.868670 




9.959769 


4.23 


10.040231 


22 


8578 


2.31 


8555 




9.960023 


4-23 


10.039977 


38 


23 


8716 


2.31 


8440 




0277 


4-23 


97 2 3 


37 


24 


8855 


2.30 


8324 




0531 


4.23 


9469 


36 


2o 


8 993 


2.30 


8209 




0784 


4.23 


9216 


35 


26 


9 J 3i 


2.30 


8093 


I.92 


1038 


4-23 


8962 


34 


27 


9269 


2.30 


7978 


I.93 


1291 


4-^3 


8709 


33 


28 


9407 


2.30 


7862 




1545 


4.23 


8455 


32 


29 


9545 


2.30 


7747 




1799 


4-^3 


8201 


31 


30 
31 


9683 


2.30 


7631 




2052 


4-^3 


7948 


30 
29 


9821 


2.29 


9.867515 




9.962306 


4-^3 


10.037694 


32 


9.829959 


2.29 


7399 




2560 


4-^3 


7440 


28 


33 


9.830097 


2.29 


7283 




2813 


4- 2 3 


7187 


27 


34 


0234 


2.29 


7167 




3067 


4.23 


6933 


26 


35 


0372 


2.29 


7051 


I.93 


3320 


4.23 


6680 


25 


36 


0509 


2.29 


6 935 


I.94 


3574 


4-^3 


6426 


24 


37 


0646 


2.29 


6819 




3827 


4-23 


6173 


23 


38 


0784 


2.29 


6703 




4081 


4- 2 3 


5919 


22 


39 


0921 


2.28 


6586 




4335 


4.23 


5665 


21 


40 
41 


1058 


2.28 


6470 




4588 


4.22 


5412 


20 
19 


9.831195 


2.28 


9.866353 


9.964842 


4.22 


10.035158 


42 


1332 


2.28 


6237 




5°95 


4.22 


4905 


18 


43 


1469 


2.28 


6120 


I.94 


5349 


4.22 


4651 


17 


44 


1606 


2.28 


6004 


I.95 


5602 


4.22 


4398 


16 


45 


1742 


2.28 


5887 




5855 


4.22 


4H5 


15 


46 


1879 


2.28 


5770 




6109 


4.22 


3891 


14 


47 


2015 


2.27 


5653 




6362 


4.22 


3638 


13 


48 


2152 


2.27 


5536 




6616 


4.22 


3384 


12 


49 


2288 


2.27 


54i9 




6869 


4.22 


3 I 3' 


11 


50 
51 


2425 


2.27 


5302 




7123 


4.22 
4.22 


2877 


10 
9 


9.832561 


2.27 


9.865185 


9.967376 


10.032624 


52 


2697 


2.27 


5068 




7629 


4.22 


2371 


8 


53 


2833 


2.27 


4950 


I.95 


7883 


4.22 


2117 


7 


54 


2969 


2.26 


4833 


I.96 


8136 


4.22 


1864 


6 


55 


3 io 5 


2.26 


4716 




8389 


4.22 


1611 


5 


56 


3 2 4i 


2.26 


4598 




8643 


4.22 


1357 


4 


57 


3377 


2.26 


4481 




8896 


4.22 


1 1 04 


3 


58 


3512 


2.26 


4363 




9149 


4.22 


0851 


2 


59 


3648 


2.26 


a ,4*45 


I.96 


9A03 
9.969656 


4.22 


0597 


1 


60 


9.833783 




9.864127 






10.030344 



M. 


Cosine. 


Diff. 1" 


Sine. 


Diff.]" 


Cotang. 


Diff. 1" 


Tang. 


132° 








47° 



84 



43° 


SINES AND TANCHSNTS. 


136° 


M. 



Sine. 


Diff. 1" 


Cosine. 


Diff.l" 


Tang. 


Diff. 1" 


Cotang. 


60 


9- 8 33783 


2.26 


9.864127 


I.96 


9.969656 


4.22 


IC.030344 


1 


3919 


2.25 


4010 


I.96 


9909 


4.22 


0091 


59 


2 


4054 


2.25 


3892 


I.97 


9.970162 


4.22 


IO.029838 


58 


3 


4189 


2.25 


3774 




0416 


4.22 


9584 


57 


4 


43*5 


2.25 


3656 




0669 


4.22 


933 1 


56 


5 


4460 


2.25 


3538 




0922 


4.22 


9078 


00 


6 


4595 


2.25 


3419 




1175 


4.22 


8825 


54 


V 


473° 


2.25 


3301 




1429 


4.22 


8571 


53 


8 


4865 


2.25 


3183 




1682 


4.22 


8318 


52 


9 


4999 


2.24 


3064 


I.97 


1935 


4.22 


8065 


51 


10 
11 


5134 
9.835269 


2.24 


2946 


I.98 


2188 


4.22 


7812 
10.027559 


50 
49 


2.24 


9.862827 


9.972441 


4.22 


12 


5403 


2.24 


2709 




2694 


4.22 


7306 


48 


13 


5538 


2.24 


2590 




2948 


4.22 


7052 


47 


14 


5672 


2.24 


2471 




3201 


4.22 


6799 


46 


15 


5807 


2.24 


2353 




3454 


4.22 


6546 


45 


16 


594i 


2.24 


2234 




3707 


4.22 


6293 


44 


17 


6075 


2.23 


2115 




3960 


4.22 


6040 


43 


18 


6209 


2.23 


1996 




4213 


4.22 


5787 


42 


19 


6343 


2.23 


1877 


I.98 


4466 


4.22 


5534 


41 


20 
21 


6477 


2.23 


1758 
9.861638 


I.99 


4719 


4.22 
4.22 


5281 


40 
39 


9.836611 


2.23 


9-974973 


10.025027 


22 


6745 


2.23 


1519 




5226 


4.22 


4774 


38 


23 


6878 


2.23 


1400 




5479 


4.22 


4521 


37 


24 


7012 


2.22 


1280 




573* 


4.22 


4268 


36 


25 


7146 


2.22 


1161 




5985 


4.22 


4015 


35 


26 


7279 


2.22 


1041 




6238 


4.22 


3762 


34 


27 


7412 


2.22 


0922 




6491 


4.22 


35°9 


33 


28 


7546 


2.22 


0802 


I.99 


6 744 


4.22 


3256 


32 


29 


7679 


2.22 


0682 


2.00 


6997 


4.22 


3003 


31 


30 
31 


7812 


2.22 


0562 
9.860442 




7250 


4.22 

4.22 


2750 
10.022497 


30 
~29~ 


9-837945 


2.22 


9-9775°3 


32 


8078 


2.21 


0322 




7756 


4.22 


2244 


2S 


33 


8211 


2.21 


0202 




8009 


4.22 


1991 


27 


34 


83441 


2.21 


9.860082 




8262 


4.22 


1738 


26 


35 


8477 


2.21 


9.859962 




8515 


4.22 


1485 


25 


36 


8610 


2.21 


9842 


2.00 


8768 


4.22 


1232 


24 


37 


8742 


2.21 


9721 


2.01 


9021 


4.22 


0979 


23 


38 


8875 


2.21 


9601 




9274 


4.22 


0726 


22 


39 


9007 


2.21 


9480 




9527 


4.22 


°473 


21 


40 
41 


9140 


2.20 
2.20 


9360 




9.979780 
9.980033 


4.22 
4.22 


10.020220 
10.019967 


20 
19 


9.839272 


9.859239 


42 


9404 


2.20 


9119 




0286 


4.22 


97H 


IS 


43 


9536 


2.20 


8998 




0538 


4.22 


9462 


17 


44 


9668 


2.20 


8877 


2.0I 


0791 


4.21 


9209 


16 


45 


9800 


2.20 


8756 


2.02 


1044 


4.21 


8956 


15 


46 


9.839932 


2.20 


8635 




1297 


4.21 


8703 


14 


47 


9.840064 


2.I9 


8514 




i55o 


4.21 


8450 


13 


48 


0196 


2.I9 


8393 




1803 


4.21 


8197 


12 


49 


0328 


2.I9 


8272 




2056 


4.21 


7944 


11 


50 


0459 


2.I9 


8151 




2309 


4-21 


7691 


10 


51 


9.840591 


2.I9 


9.858029 




9.982562 


4.21 


10.017438 


9 


52 


0722 


2.I9 


7908 




2814 


4.21 


7186 


S 


53 


0854 


2.I9 


7786 


2.02 


3067 


4.21 


6933 


1 


54 


0985 


2.I9 


7665 


2.03 


3320 


4.21 


6680 


6 


55 


1 1 in 


2.I9 


7543 




3573 


4.21 


6427 


° 


56 


1247 


2.l8 


7422 




3826 


4.21 


6174 


4 | 


57 


1378 


2.l8 


7300 




4079 


4.21 


5921 


3 1 


58 


1509 


2.l8 


7178 




433 1 


4.21 


5669 


2 1 


59 


1640 


2.l8 


7056 


2.O3 


4584 


4.21 


5416 


1 1 


60 


9.841771 
Cosine. 




9.856934 

Sine. 


Diff.l" 


9.984837 




Diff. 1" 


10.015163 
Tang. 


(i 
M. 


Diff. 1" 


Cotang. 


133° 












46° 



27 



85 



44° 



ZiOaARITHMXC 



131 



M. 

~ 

1 
2 
3 
4 
5 

6 
7 
8 
9 
10 

TT 

12 
13 
14 
15 

16 
17 
18 
19 
20 

"21 

22 
23 

24 
25 

26 
27 
28 
29 
J50 

IT 
32 
33 
34 

35 

36 
37 
38 
39 
40 

~41 
42 
43 
44 
45 

46 
47 
48 
49 
50 

~5T 
52 
53 

54 
55 

56 
57 
58 
59 
60 



Sine. 



I)iif. 1" 



9.841771 


2.18 


1902 


2.18 


2033 


2.18 


2163 


2.17 


2294 


2.17 


2424 


2.17 


2555 


2.17 


2685 


2.17 


2815 


2.17 


2946 


2.17 


3076 


2.17 


9.843206 


2.16 


333 6 


2.16 


3466 


2.16 


3595 


2.16 


3725 


2.16 


3855 


2.16 


3984 


2.16 


4114 


2.16 


4 2 43 


2.15 


4372 


2.15 


9.844502 


2.15 


4631 


2.15 


4760 


2.15 


4889 


2.15 


5018 


2.15 


5H7 


2.15 


5276 


2.14 


5405 


2.14 


5533 


2.14 


5662 


2.14 


9.845790 


2.14 


59i9 


2.14 


6047 


2.14 


6175 


2.14 


6304 


2.14 


6432 


2.13 


6560 


2.13 


6688 


2.13 


6816 


2.13 


6944 


2.13 


9.847071 


2.13 


7199 


2.13 


73 2 7 


2.13 


7454 


2.12 


7582 


2.12 


7709 


2.12 


7836 


2.12 


7964 


2.12 


8091 


2.12 


8218 


2.12 


9.848345 


2.12 


8472 


2. II 


8599 


2. II 


.8726 


2. II 


8852 


2. II 


8979 


2. 1 1 


9106 


2. II 


9232 


2. II 


9359 


2. II 


9.849485 




Cosine. 


Diff. 1" 



Cosine. 

9.856934 
6812 
6690 
6568 
6446 
6323 

6201 
6078 
5956 
5833 
57H 



Diff. 1" 



9.855588 
5465 
534 2 
5219 
5096 

4973 
4850 
4727 
4603 
4480 



9.854356 

4 2 33 
4109 
3986 
3862 

3738 
3614 

349° 
3366 
3242 



9.853118 
2994 
2869 

2 745 
2620 

2496 
2371 
2247 
2122 
1997 



9.851872 
1747 
1622 
1497 
1372 

1246 
1121 

0996 
0870 

0745 



9.850619 
0493 
0368 
0242 

9.8501 16 

9.849990 
9864 

9738 
9611 

9-849485 

Sine. 



2.04 
2.05 



2.05 
2.06 



2.06 
2.07 



2.07 
2.08 



2.08 
2.09 



2.09 
2.IO 



Diff.1' 



Tang. 

9.984837 
5090 

5343 
5596 
5848 
6101 

6354 
6607 
6860 
7112 
73 6 5 



9.987618 
7871 
8123 
8376 
8629 

8882 

9*34 

9387 

9640 

9.989893 



9.99014c 
0398 
0651 
0903 
1156 

1409 

1662 
1914 
2167 
2420 



9.992672 
2925 

3178 

343° 
3683 

393 6 
4189 

444 1 
4694 

4947 



9.995199 

545 2 
5705 
5957 
6210 

6463 
6715 
6968 
7221 
7473 



9.997726 

7979 
8231 

8484 

8737 

8989 

9242 

9495 

9-999747 

10.000000 

Cotang. 



Diff. 1" 

4.2 

4.2 
4.2 
4.2 
4.2 
4.2 

4.2 
4.2 
4.2 
4.2 
4.2 



4.2 
4.2 
4.2 
4.2 
4.2 

4.2 
4.2 
4.2 
4.2 
4.2 



4.2 
4.2 
4.2 
4.2 
4.2 

4.2 

4.2 
4.2 
4.2 
4.2 



4.2 
4.2 
4.2 
4.2 
4.2 

4.2 
4.2 
4.2 
4.2 
4.2 



4.2 
4.2 
4.2 
4.2 

4.2 

4.2 
4.2 
4.2 
4.2 
Ar% 

4.2 
4.2 
4.2 
4.2 
4.2 

4.2 
4.2 
4.2 

4.2 



Diff. 1" 



Cotang. 

10.015163 
4910 
4657 
4404 
4152 
3899 
3646 

3393 
3140 
2888 
2635 



10.012382 
2129 
1877 
1624 
i37i 
1118 
0866 
0613 
0360 

10.010107 



10.009855 
9602 

9349 
9097 
8844 

8591 

8338 
8086 

7833 

758o 



10.007328 

7075 

6822 

6570 

6 3 r 7 

6064 

5811 

5559 
5306 

5°53 



10.004801 

4548 
4295 
4043 

379° 

3537 
3285 

3032 
2779 
2527 



10.002274 
2021 
1769 
1516 
1263 

ion 

0758 

0505 

0253 

10.000000 



Tang. 



60 
59 
58 
57 
56 
55 

54 
53 
52 
51 
50 

~W 
48 
47 
46 
45 

44 
43 
42 
41 
40 



39 
38 
37 
36 
35 

34 
33 
32 
31 

30 



29 
28 
27 
26 
25 

24 

23 
22 
21 
20 

~19~ 
18 
17 
16 
15 

14 
13 
12 
11 

10 



5 

4 
3 

2 
1 


M. 



134 c 



45 c 



86 



TABLE 



OF 



NATURAL SINES 



AND 



COSINES. 



8? 



STATURAL SINES AND COSINES. 




/ 



1 

2 
3 
4 
5 

6 
7 
8 
9 
10 

11 

12 
13 
14 
15 

16 
17 

18 
19 
20 

21 

22 
23 
24 
25 

26 

27 
28 
29 
30 

31 

32 
33 
34 
35 

36 
37 
38 
39 
40 

41 

42 
43 
44 
45 

46 

47 
48 
49 
50 

~5~f 
52 
53 
54 
55 

56 
57 
58 
59 
60 

/ 


0° 


1° 


2° 


3° 


4° 


/ 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 

44 

43 
42 
41 
40 

39 
38 
37 
36 
35 

34 
33 
32 
31 
30 

29 
28 
27 
26 
25 

24 
23 
22 
21 
20 

19 
18 
17 
16 
15 

14 
13 
12 
11 
10 

9 
8 
7 
6 
5 

4 
3 
2 
1 


/ 




Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 




OOOOO 

00029 
00058 
00087 
00116 
00145 

00175 
00204 
00233 
00262 
00291 

00320 
00349 
00378 
00407 
00436 

00465 

00495 
00524 

00553 
00582 


Unit. 
Unit. 
Unit. 
Unit. 
Unit. 
Unit. 

Unit. 
Unit. 
Unit. 
Unit. 
Unit. 

99999 
99999 
99999 
99999 
99999 

99999 
99999 
99999 
99998 
99998 

99998 
99998 
99998 
99998 
99997 
99997 
99997 
99997 
99996 
99996 

99996 
99996 
99995 
99995 
99995 

99995 
99994 
99994 
99994 
99993 

99993 
99993 
99992 
99992 
99991 

99991 
99991 
99990 
99990 
99989 
99989 

99989 
99988 
99988 
99987 

99987 
99986 
99986 
99985 
99985 

Sine. 


01745 
01774 
01803 
01832 
01862 
01891 

01920 
01949 
01978 
02007 
02036 

02065 
02094 
02123 
02152 
02181 

0221 1 

02240 
02269 
02298 
02327 

02356 
02385 
02414 
02443 
02472 

02501 
02530 
02560 
02589 
02618 

02647 
02676 
02705 

02734 
02763 

02792 
02821 
02850 
02879 
02908 


99985 
99984 
99984 
99983 
99983 
99982 

99982 
99981 
99980 
99980 
99979 


03490 

03519 
03548 

03577 
03606 

03635 
03664 

03693 
03723 
03752 
03781 

03810 

03839 
03868 

03897 
03926 

03955 
03984 
04013 
04042 

04071 


99939 
99938 
99937 
99936 
99935 
99934 

99933 
99932 

9993 1 
9993° 
99929 


05234 
05263 
05292 
05321 

05350 
05379 

05408 

05437 
05466 

05495 
05524 

05553 
05582 
05611 
05640 
05669 

05698 
05727 
05756 

05785 
05814 


99863 
99861 
99860 
99858 
99857 
99855 

99854 
99852 
99851 

99849 
99847 


06976 
07005 

07034 
07063 
07092 
07121 

07150 
07179 
07208 
07237 
07266 

07295 
07324 

07353 
07382 

0741 1 

07440 
07469 
07498 
07527 

07556 


99756 

99754 
99752 
99750 
99748 
99746 

99744 
99742 

9974° 
99738 
99736 

99734 
99731 

99729 
99727 
99725 

99723 
99721 
99719 
99716 
99714 




99979 
99978 
99977 
99977 
99976 

99976 
99975 
99974 
99974 
99973 


99927 
99926 

99925 
99924 
99923 

99922 

999*1 
99919 
99918 
99917 


99846 

99844 
99842 
99841 

99839 

99838 
99836 

99834 
99833 
99831 

99829 
99827 
99826 
99824 
99822 

99821 
99819 
99817 
99815 

99813 
99812 
99810 
99808 
99806 
99804 

99803 
99801 

99799 
99797 
99795 

99793 
99792 
99790 
99788 
99786 

99784 
99782 
99780 
99778 
99776 

99774 
99772 
9977° 
99768 
99766 

99764 
99762 
99760 
99758 
99756 

Sine. 




00611 
00640 
00669 
00698 
00727 

00756 
00785 
00814 
00844 
00873 


99972 
99972 
99971 

99970 
99969 

99969 
99968 

99967 
99966 
99966 

99965 
99964 
99963 
99963 
99962 

99961 
99960 

99959 
99959 
99958 

99957 
99956 
99955 
99954 
99953 
99952 
99952 

99951 
99950 

99949 
99948 
99947 
99946 
99945 
99944 

99943 
99942 
99941 

9994° 
99939 

Sine. 


04100 
04129 
04159 
04188 
04217 

04246 

04275 

04304 

04333 
04362 


99916 
999 J 5 
999*3 
99912 

99911 

99910 
99909 
99907 
99906 
99905 

99904 
99902 
99901 
99900 
99898 

99897 
99896 

99894 
99893 
99892 


05844 

05873 
05902 

0593 1 
05960 

05989 
06018 
06047 
06076 
06105 


07585 
07614 
07643 
07672 
07701 

07730 

07759 
07788 
07817 
07846 

07875 
07904 

07933 
07962 
07991 

08020 
08049 
08078 
08107 
08136 

08165 
08194 
08223 
08252 
08281 

08310 

08339 
08368 
08397 
08426 

o8455 
08484 

08513 
08542 
08571 

08600 
08629 
08658 
08687 
08716 


99712 
99710 
997o8 

99705 
99703 

99701 

99699 
99696 

99694 
99692 




00902 
00931 
00960 
00989 
01018 

01047 
01076 
01105 
01134 
01164 


04391 

04420 

04449 
04478 
04507 
04536 

04565 
04594 
04623 

04653 


06134 
06163 
06192 
06221 
06250 

06279 
06308 
06337 
06366 
06395 
06424 
06453 
06482 
06511 
06540 

06569 
06598 
06627 
06656 
06685 

06714 

o6743 
06773 
06802 
06831 

06860 
06889 
06918 

06947 
06976 


99689 

99687 
99685 
99683 
99680 

99678 
99676 

99673 
99671 

99668 




01193 
01222 
01251 
01280 
01309 

01338 
01367 
01396 
01425 
OI454 
01483 
01513 
01542 
01571 
01600 

01629 
01658 
01687 
01716 
01745 

Cosine 


02938 
02967 
02996 
03025 
03054 
03083 
03112 
03141 
03170 
03199 

03228 

03257 
03286 
03316 
03345 

03374 
03403 

03432 
03461 

03490 


04682 
047 1 1 
04740 

04769 
04798 

04827 
04856 
04885 
04914 
04943 

04972 
05001 
05030 
05059 
05088 

05H7 

05146 
05175 
05205 
05234 


99890 
99889 
99888 
99886 
99885 

99883 
99882 
99881 
99879 
99878 

99876 
99875 
99873 
99872 
99870 

99869 
99867 
99866 
99864 
99863 

Sine. 


99666 
99664 
99661 

99659 
99657 

99654 
99652 
99649 

99647 
99644 

99642 
99639 
99637 
99635 
99632 

99630 
99627 
99625 
99622 
99619 

Sine. 




Cosine. 


Cosine. 


Cosine. 


Cosine. 




89° 


88° 


87° 


86° 


85° 





NATURAL SINES AND COSINES. 




/ 



i 

2 
3 

4 
5 

6 
7 
8 
9 
10 

11 

12 
13 
14 
15 

16 
17 

18 
19 
20 

21 
22 
23 
24 
25 

26 

27 
28 
29 
30 

31 
32 

oo 

34 
35 

36 
37 
38 
39 

I 40 

41 
42 

43 
44 
45 

46 

47 
48 
49 
50 

51 
52 
53 
54 
55 

56 
57 
58 
59 
60 

/ 


5° 


6° 


7 o 


8° 


9 b 


/ 

60 

59 

58 1 

57 

56 

55 

54 
53 
52 
51 
50 

~49~ 
48 
47 
46 
45 

44 
43 
42 
41 
40 

39 
38 
37 
36 
35 

34 
33 
32 
31 
30 

29 

28 
27 
26 
25 

24 
23 
22 
21 
20 

19 
18 
17 
16 
15 

14 
13 
12 
11 
10 

9 

8 
7 
6 
5 

4 
3 
2 
1 


/ 




Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 




08716 
08745 
08774 
08803 
08831 
08860 

08889 
08918 
08947 
08976 
09005 


99619 
99617 
99614 
99612 
99609 
99607 

99604 
99602 

99599 
99596 
99594 


10453 
10482 
10511 
10540 
10569 
10597 

10626 
10655 
10684 
IO713 

10742 

10771 
10800 
10829 
10858 
10887 

10916 
IO945 
10973 
1 1002 
II031 

II060 
II089 
IIIl8 
1 1 147 
III76 

II205 
II234 
II263 
II291 

II320 


99452 
99449 

99446 
99443 
99440 
99437 

99434 
99431 
99428 

99424 
99421 
99418 

99415 
99412 

99409 
99406 

99402 
99399 
99396 
99393 
9939° 
99386 
99383 
99380 
99377 
99374 
99370 
99367 
99364 
99360 

99357 

99354 
99351 
99347 
99344 
99341 

99337 
99334 
9933 1 
99327 
99324 
99320 
99317 

993*4 
99310 

99307 
99303 

99300 
99297 

99293 
99290 

99286 

99283 

99279 
99276 

99272 
99269 

99265 
99262 
99258 
99255 

Sine. 


12187 
12216 
12245 
12274 
12302 
12331 

12360 
12389 
12418 

12447 
12476 


99255 
99251 
99248 
99244 
99240 

99237 

99233 
99230 
99226 
99222 
99219 
99215 
99211 
99208 
99204 
99200 

99197 

99193 
99189 
99186 
99182 

99178 

99175 
99171 
99167 
99163 
99160 
99156 
99152 
99M8 
99H4 

99I4I 
99137 
99133 
99129 
99125 
99122 
99118 

99114 

99110 
99106 


13917 
13946 

13975 
14004 

14033 
I4061 

14090 

I4II9 

14148 

I4I77 
14205 


99027 
99023 
99019 
99015 
99011 
99006 

99002 
98998 

98994 
98990 
98986 

98982 

98978 
98973 
98969 

98965 

98961 

98957 

98953 
98948 

98944 

98940 

98936 

98931 
98927 
98923 

98919 
98914 
98910 
98906 
98902 

98897 
98893 
98889 
98884 
98880 

98876 
98871 
98867 
98863 
98858 

98854 
98849 

98845 
98841 
98836 

98832 
98827 
98823 
98818 
98814 

98809 
98805 
98800 

98796 
98791 

98787 
98782 
98778 
98773 
98769 


15643 
15672 

15701 
15730 
15758 
15787 

15816 

15845 

15873 
15902 

15931 


98769 
98764 
98760 

98755 
98751 
98746 

98741 
98737 

98732 
98728 

98723 

98718 

98714 
98709 
98704 
98700 

98695 
98690 
98686 
98681 
98676 




09034 
09063 
09092 
09121 
09150 

09179 
09208 

09237 
09266 

09295 
09324 

09353 
09382 
0941 1 
09440 

09469 
09498 

09527 
09556 
09585 

09614 
09642 
09671 
09700 
09729 

09758 
09787 
09816 
09845 

09874 

09903 
09932 
09961 
09990 
10019 

10048 
IOO77 
IOI06 
IOI35 
IO164 


9959 1 
99588 
99586 
99583 
99580 

99578 
99575 
99572 
99570 

99567 


12504 

12533 
12562 
12591 
12620 

12649 
12678 
12706 

12735 
12764 


14234 
14263 
14292 
14320 
14349 
14378 
14407 
14436 
14464 
14493 


15959 
15988 
16017 
16046 
16074 

16103 
1 6 1 3 2 
16160 
16189 
16218 

16246 
16275 
16304 
16333 
16361 

16390 

16419 

16447 
16476 

16505 




99564 
99562 

99559 
99556 
99553 | 

99551 
99548 

99545 
99542 
99540 

99537 
99534 
9953 1 
99528 
99526 

99523 
99520 

99517 
995H 
99511 

99508 
99506 

99503 
99500 

99497 

99494 
99491 
99488 
99485 
99482 


12793 
12822 
12851 
12880 
12908 

12937 
12966 
12995 
13024 

13053 


14522 

14551 
14580 

14608 

14637 

14666 

14695 

14723 
14752 
I478I 


98671 
98667 
98662 
98657 
98652 

98648 
98643 
98638 
98633 
98629 




1 1 349 

II378 
1 1407 
II436 
II465 

1 1 494 
II523 

11552 
11580 
11609 

11638 
11667 
11696 
II725 

"754 

11783 
11812 
1 1 840 
11869 
11898 

11927 
11956 
1 1985 
12014 
12043 

12071 
12100 
12129 
12158 
12187 


13081 
13HO 
13139 
13168 
13197 
13226 

13254 
13283 

I33I2 
13341 
13370 
13399 

13427 
13456 
13485 

i35 J 4 

13543 
13572 

13600 

13629 

13658 
13687 
13716 
13744 
13773 
13802 

13831 
13860 
13889 
13917 


14810 
14838 
14867 
14896 

14925 

14954 
14982 
15011 

15040 
15069 


16533 
16562 

16591 
16620 
16648 

16677 
16706 

16734 
16763 
16792 


98624 
98619 
98614 
98609 
98604 

98600 

98595 
98590 

98585 
98580 

98575 
98570 

98565 
98561 

98556 

98551 
98546 
98541 

98536 
98531 

98526 
98521 
98516 

985H 
98506 

98501 
98496 

98491 
98486 
98481 

Sine. 




99102 
99098 
99094 

99091 
99087 

99083 
99079 

99075 
99071 
99067 

99063 

99059 
99055 
99051 
99047 

99043 
99039 

99035 
99031 
99027 

1 Sine. 


15097 
15126 

15155 
15184 
15212 

15241 

15270 

15299 
15327 
15356 


16820 
16849 
16878 
16906 
16935 
16964 
16992 
17021 
17050 
17078 




IOI92 
I022I 
IO25O 
IO279 
IO308 

10337 
IO366 
IO395 
IO424 

10453 


99479 
99476 
99473 
9947o 
99467 

99464 
99461 

99458 

99455 
99452 


15385 
I54H 
15442 

15471 
15500 

15529 

15557 
15586 

I56I5 
15643 


17107 
17136 
17164 

17193 

17222 

17250 

17279 
17308 

17336 

17365 

Cosine. 




Cosine. 


Sine. 


Cosine. 


Cosine. 


Cosine. 


Sine. 




84° 


83° 


82° 


81° 


80° 





89 



NATURAL SIBTES AND COSINES. 


/ 



1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 

20 

21 

22 
23 
24 
25 

26 
27 
28 
29 
30 

31 
32 
33 
34 
35 

36 

37 
38 
39 
40 

41 
42 
43 
44 
45 

46 

47 
48 
49 
50 

1 51 
52 

53 

54 
55 

56 
57 
58 
59 
60 

/ 


10° 


11° 


12° 


13° 


14° 


/ 

60 
59 
58 
57 
56 
55 

54 
53 
52 
51 

50 

49 
48 
47 
46 
45 

44 
43 
42 
41 
40 

39 
38 
37 
36 
35 

34 
33 
32 
31 

30 

29 
28 
27 
26 
25 

24 
23 

22 
21 
20 

19 
18 
17 
16 
15 

14 
13 
12 
11 
10 

~!T 

8 
7 
6 
5 

4 
3 
2 
1 


/ 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


17365 
17393 
17422 
1 745 1 
17479 
17508 

17537 
17565 
17594 
17623 
17651 


98481 
98476 
98471 
98466 
98461 
98455 
98450 

98445 
98440 

98435 
98430 

98425 
98420 
98414 
98409 
98404 

98399 
98394 
98389 

98383 
98378 

98373 
98368 
98362 

98357 
98352 

98347 
98341 

98336 
98331 
98325 
98320 

98315 
98310 
98304 
98299 

98294 
98288 
98283 
98277 
98*72 

98267 
98261 
98256 
98250 
98245 

98240 

98234 
98229 
98223 
98218 

98212 
98207 
98201 
98196 
98190 

98185 
98179 
98174 
98168 
98163 

Sine. 


19081 
19109 
19138 
19167 
19195 
I9224 

19252 
19281 
19309 
19338 
19366 


98163 
98157 
98152 
98146 
98140 
98135 

98129 
98124 
98118 
98112 
98107 

98101 
98096 
98090 
98084 
98079 

98073 
98067 

98061 
98056 
98050 


20791 
20820 
20848 
20877 
20905 
20933 
20962 
20990 
21019 
21047 
21076 

21104 
21132 
21161 
21189 
21218 

21246 
21275 
21303 

2I33 1 

21360 

21388 

21417 
21445 
21474 
21502 

21530 
21559 
21587 
21616 
21644 

21672 
21701 
21729 
21758 
21786 

21814 
21843 
21871 
21899 
21928 

21956 
21985 
22013 
22041 
22070 

22098 
22126 

22155 
22183 
22212 

22240 
22268 
22297 
22325 
22353 

22382 
22410 
22438 
22467 
22495 

Cosine. 


97815 
97809 
97803 

97797 
97791 

97784 

97778 
97772 
97766 
97760 
97754 
97748 
97742 
97735 
97729 
97723 
97717 
97711 

97705 
97698 
97692 

97686 
97680 

97673 
97667 
97661 

97655 
97648 
97642 
97636 
97630 

97623 
97617 
97611 
97604 
97598 
97592 
97585 
97579 

97573 
97566 

97560 

97553 
97547 
97541 
97534 
97528 
97521 

97515 
97508 

97502 
97496 
97489 
97483 
97476 

97470 

974 6 3 

97457 
9745° 
97444 
97437 

Sine. 


22495 
22523 
22552 
22580 
22608 
22637 

22665 
22693 
22722 
22750 
22778 

22807 
22835 
22863 
22892 
22920 

22948 
22977 

23005 
23033 
23062 

23090 
23118 
23146 

23 J 75 
23203 

23231 

23260 
23288 
23316 
23345 


97437 
9743° 
97424 
97417 
97411 

97404 
97398 
97391 
97384 
97378 
97371 

97365 
97358 
97351 
97345 
97338 

97331 

97325 
97318 

973 11 
97304 


24192 
24220 
24249 
24277 

24305 
24333 

24362 
24390 
24418 
24446 
24474 

24503 
24531 
24559 
24587 
24615 

24644 
24672 
24700 
24728 
24756 


97°3° 
97023 
97015 
97008 
97001 
96994 

96987 
96980 

96973 
96966 
96959 


17680 
17708 

17737 
17766 

17794 
17823 
17852 
17880 
17909 
17937 


19395 
19423 
19452 
1 948 1 

I95°9 

19538 
19566 

19595 
19623 

19652 


96952 
96945 
96937 
96930 
96923 

96916 
96909 
96902 
96894 
96887 


17966 

17995 
18023 
18052 
18081 

1 8 109 

18138 
18166 

18195 
18224 

18252 
18281 
18309 

18338 
18367 

18395 
18424 
18452 
18481 
18509 


19680 
I9709 
19737 
19766 
I9794 

19823 
19851 
19880 
19908 
19937 
19965 
19994 
20022 
20051 
20079 

20108 
20136 
20165 
20193 
20222 


98044 
98039 
98033 
98027 
98021 

98016 
98010 
98004 
97998 
97992 

97987 
97981 

97975 
97969 
97963 

97958 
97952 
97946 

9794° 
97934 

97928 
97922 
97916 
97910 
97905 
97899 

97893 
97887 
97881 

97875 
97869 
97863 

97857 
97851 

97845 

97839 
97833 
97827 
97821 

978i5 
Sine. 


97298 
97291 

97284 
97278 
97271 

97264 
97257 
9725 1 
97244 
97237 
97230 
97223 
97217 
97210 
97203 

97196 
97189 
97182 
97176 
97169 

97162 

97155 
97148 
97141 
97134 

97127 
97120 

97"3 

97106 
97100 

97093 
97086 

97079 
97072 

97065 

97058 
97051 

97044 
97037 
97030 

Sine. 


24784 
24813 
24841 
24869 
24897 

24925 

24954 
24982 
25010 
25038 


96880 

96873 
96866 
96858 
96851 

96844 
96837 
96829 
96822 
96815 

96807 
96800 
96793 
96786 
96778 

96771 

96764 
96756 
96749 
96742 

96734 
96727 
96719 
96712 
96705 
96697 
96690 
96682 

96675 
96667 

96660 
96653 

96645 
96638 

96630 

96623 
96615 
96608 
96600 
96593 

Sine. 


23373 
23401 

23429 
23458 
23486 

23514 
23542 
23571 

23599 

23627 

23656 
23684 
23712 

23740 
23769 

23797 
23825 

23853 
23882 
23910 

23938 
23966 

23995 
24023 
24051 

24079 
24108 
24136 
24164 
24192 


25066 
25094 
25122 
25151 
25179 

25207 

25235 
25263 
25291 
25320 


18538 
18567 
18595 
18624 
18652 

18681 
18710 
18738 
18767 
18795 


20250 
20279 
20307 
20336 
20364 

20393 
20421 
20450 
20478 
20507 

20535 
20563 
20592 
20620 
20649 

20677 
20706 

20734 
20763 
20791 


25348 
25376 

25404 
25432 
25460 

25488 
25516 
25545 
25573 
25601 

25629 
25657 
25685 
25713 
25741 

25769 
25798 
25826 
25854 
25882 


18824 
18852 
18881 
18910 
18938 

18967 
18995 
19024 
19052 
19081 

Cosine. 


Cosine. 


Cosine 


Cosine. 


79° 


78° 


77° 


I 76 o 

1 


75° 



90 



NATURAL SIN US AND COSINES. 


ho 


15° 

I 


16° 


17° 


18° 


19° 


/ 

60 


Sine. 


Cosine, j 


Sine. 


Cosine. 


Sine. 


1 

Cosine, j 


Sine. 


Cosine. 


Sine. 


Cosine. 


25882 


9 6 593 ! 


27564 


96126 


29237 


95630 s 


30902 


95106! 


32557 


94552 


j 


25910 


96585 


27592 


96118 


29265 


95622 


30929 


95097 ! 


32584 


94542 


59 


i 2 


25938 


96578 


27620 


96110 


29293 


95613 ! 


30957 


95088! 


32612 


94533 


58 1 


i 6 


25966 


96570 


27648 


96102 


29321 


95605: 


30985 


95079! 


32639 


94523 


57 1 


4 


25994 


96562 


27676 


96094 


29348 


95596; 


3101a 


95070 


32667 


945*4 


56 





26022 


9 6 555 i 


27704 


96086 


29376 


95588 


3*040 


95061 : 


32694 


94504 


00 


6 


26050 


9 6 547 1 


27731 


96078 


29404 


95579 


31068 


95052 | 


32722 


94495 


54 


7 


26079 


96540 


27759 


96070 j 


29432 


9557i 


31095 


95043 ; 


32749 


94485 


53 


8 


26107 


96532 


27787 


96062 : 


29460 


95562 


3**23 


95033! 


i 32-777 


94476 


52 


9 


26135 


96524! 


27815 


96054' 


29487 


95554 


31151 


95024 j 


,32804 


94466 


51 


10 
11 


26163 
26191 


96517: 
96509' 


27843 
27871 


96046 
96037 I 


29515 
29543 


95545 
95536 


31*78 


95015, 

95006 1 


; 32832 
32859 


94457 
94447 


50 
49 


31206 


12 


26219 


96502 


27899 


96029 


29571 


95528 


31233 


94997; 


j 32887 


94438 


48 


13 


26247 


96494 


27927 


96021 


29599 


95519 J 


3*26i 


94988, 


329*4 


94428 


47 


14 


26275 


96486 j 


2 7955 


96013 


29626 


955" 


31289 


94979: 


! 32942 


944*8 


46 


15 


26303 


96479 


27983 


96005 


29654 


95502 


3*3*6 


94970; 


32969 


94409 


45 


16 


26331 


96471 


28011 


95997 


29682 


95493 


3*344 


94961 1 


!32997 


94399 


44 


IV 


26359 


96463! 


28039 


959 8 9 


29710 


95485| 


3*372 


94952 


33024 


94390 


43 


18 


26387 


96456 


28067 


95981! 


29737 


95476, 


3*399 


94943 ' 


i 33051 


9438o 


42 | 


19 


26415 


96448 | 


28095 


9S97 2 -; 


29765 


95467 


31427 


94933! 


i 33079 


9437° 


41 


20 
~~21 


26443 
26471 


96440 
9 6 433 


28123 
28150 


95964J 
9595 6 l 


29793 
29821 


95459 
9545° 


3*454 


94924! 
949*5; 


33106 
! 33*34 


94361 
9435* 


40 
39 


31482 


22 


26500 


96425 


28178 


95948 


29849 


95441 


!3*5*° 


94906 ] 


33161 


94342 


38 


23 


26528 


96417 


28206 


95940 j 


29876 


95433! 


13*537 


94897! 


j33* 8 9 


94332 


37 


24 


26556 


96410 


28234 


9593* 


29904 


95424| 


i3*5 6 5 


94888: 


33216 


94322 


36 


2o 


26584 


96402 


28262 


959 2 3 


29932 


954i5i 


:3*593 


94878! 


33244 


943*3 


35 


26 


26612 


96394 


28290 


959 r 5 ! 


29960 


95407 


31620 


94869 


1 33271 


94303 


34 


2Y 


26640 


96386 


28318 


95907 ; 


29987 


95398! 


13*648 


94860; 


I33298 


94293 


33 


28 


26668 


96379 


28346 


95898 j 


30015 


95389 


3*675 


94851 , 


'33326 


94284 


32 


29 


26696 


96371 


^374 


95890, 


30043 


9538o 


3*703 


94842, 


'33353 


94274 


31 


30 


26724 


96363 


28402 


95882; 


30071 


95372 


3*73° 


94832! 


3338i 


94264 


30 


31 


26752 


9 6 355 


28429 


95874 


30098 


95363 


31758 


94823! 


! 33408 


94254 


29 


32 


26780 


9 6 347 


28457 


95865 


30126 


95354 


31786 


94814' 


i 33436 


94245 


28 


33 


26808 


96340 


28485 


95857 


30154 


95345 


3*8i3 


94805, 


1 33463' 


942 35 


27 


34 


26836 


96332 


28513 


95849! 


30182 


95337 1 


I31841 


94795' 


3349° 


94225 


26 


35 


26864 


96324 


28541 


9584! 


30209 


95328 


1 31868 


94786! 


1335*8 


94215 


25 


36 


26892 


96316 


28569 


95832: 


30237 


953^! 


J31896 


94777 ; 


! 33545 


94206 


24 


37 


26920 


96308 


28597 


95824! 


30265 


95310! 


!3*9 2 3 


94768 


133573 


94*96 


23 


38 


26948 


96301 


28625 


95816; 


30292 


953oi i 


i3*95* 


94758 


1 33600 


94186 


22 


39 


26976 


96293 


28652 


95807 : 


30320 


95293; 


(3*979 


94749 


'33627 


94*76 


21 


40 


27004 


96285 


28680 


95799 i 


30348 


95284 


i 12OO'0 


94740 | 


'33655 


94*67 


20 


41 


27032 


96277 


28708 


9579ij 


30376 


95275 


I32034 


94730 | 


! 33682 


94*57 


19 


42 


27060 


96269 


28736 


95782! 


30403 


95266 ; 


32061 


94721 


33710 


94*47 


IS 


43 


27088 


96261 


28764 


95774 


3°43* 


95257 


j 32089 


947*2 


'33737 


94*37 


17 


44 


27116 


96253 


28792 


95766, 


30459 


95248; 


! 32116 


94702 


33764 


94*27 


16 


46 


27144 


96246 


28820 


95757 


30486 


95 2 4° ! 


! 32144 


94693 


,33792 


941 1 8 


15 


46 


27172 


96238 


28847 


95749! 


30514 


95231 1 


J32I7I 


94684' 


33819 


94108 


14 


47 


27200 


96230 


28875 


9574° 1 


30542 


95222 


[32199 


94 6 74 


33846 


94098 


13 


48 


27228 


96222 


28903 


95732; 


3°57° 


95213' 


1 32227 


94665 


33874 


94088 


12 


49 


27256 


96214 j 


28931 


95724! 


3°597 


95204 


! 32254 


94656 


33901 


94078 


I* 


50 


27284 


96206 


j 28959 


95715; 


30625 


95*95 


* 32282 


94646 


' 33929 


94068 


10 


51 


27312 


96198 


28987 


95707! 


30653 


95186' 


'32309 


94637 


! 3395 6 


94058 


9 


52 


27340 


96190 


i 29015 


95698 


30680 


95*77 


32337 


94627 


33983 


94049 


8 


53 


27368 


96182 


'29042 


95690 


3070S 


95168 


32304 


94618 


3401 1 


94039 


7 


54 


27396 


96174 


1 29070 


95681 


30736 


95*59 ! 


32392 


94609 


34038 


94029 


6 


55 


27424 


96166 


29098 


95 6 73| 


30763 


95150 


,32419 


94599 


34065 


940*9 


5 


56 


27452 


96158 


29126 


95664! 


30791 


95142 


'32447 


9459° 


34093 


94009 


4 


57 


27480 


96150 


29154 


95656 


30819 


95*33. 


! 32474 


94580 


34120 


93999 


3 i 


58 


27508 


96142 


29182 


95647' 


30846 


95124 


32502 


9457* 


34*47 


93989 


2 i 


59 


27536 


96134 


1 29209 


95639: 


30874 


95**5 


32529 


94561 


34*75 


93979 


1 | 


j 60 
/ 


27564 
Cosine. 


96126 
Sine. 


29237 


95 6 3° , 


30902 


95106 


32557 


94552 


,34202 
Cosine. 

: 


93969 

Sine. 




/ 
— =i 


1 Cosine. 

I 


Sine, j 


Cosine. 


1 Sine. 


Cosine. 


Sine 


74° 


73° 


72° 


71° 


70° 



91 



NATURAL SINES AND COSINES. 


/ 



20° 21° 


22° 


23° 


24° 


/ 

60 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 
37461 


Cosine. 


39073 


Cosine. 


£ine. 


Cosine. 


34202 


93969 


35837 


93358 


92718 


92050 


40674 


9*355 


1 


34229 


93959 , 


358H 


93348 


37488 


92707 


39100 


92039 


40700 


9*343 


59 


2 


34257 


93949: 


35891 


93337 


37515 


92697 


39127 


92028 


40727 


91331 


58 




34284 


93939 


359 l8 


93327 


37542 


92686; 


39153 


92016 


40753 


9*3*9 


57 


4 


343 11 


93929 


35945 


93316 


37569 


92675 I 


39180 


92005 


40780 


9*307 


56 


5 


34339 


939*9, 


35973 


93306 


37595 


92664' 


39207 


91994 


40806 


9*295 


55 


6 


343 66 


93909 


36000 


93295 


37622 


92653 | 


39234 


91982 


40833 


91283 


54 


i 


34393 


938991 


36027 


93285 


37649 


92642 


39260 


91971 


40860 


91272 


53 


8 


34421 


93889! 


36054 


93274 


37676 


92631 


39287 


9*959 


40886 


91260 


52 


9 


3444 8 


93879 


36081 


93264 


37703 


92620 1 


39314 


91948 


409*3 


91248 


51 


10 

11 


34475 
345°3 


93869 
93859 


36108 
36135 


93253 


37730 
37757 


92609 
92598 


3934i 
39367 


9*936 
91925 


40939 
40966 


91236 


50 
49 


93243 


9*224 


12 


34530 


93849 


36162 


93232 


37784 


92587 


39394 


91914 


40992 


91212 


48 


13 


34557 


93839 


36190 


93222 


37 o X o 


92576 


39421 


91902 


4*0*9 


91200 


47 


U 


345H 


93829 


36217. 


93211 


37838 


92565 


39448 


91891 


4*045 


91188 


46 


15 


34612 


93819 


36244 


93201 


37865 


92554 


39474 


91879 


41072 


91176 


45 


16 


34639 


93809 


36271 


93190 


37892 


92543 


395 QI 


91868 


41098 


91164 


44 


17 


34666 


93799 


36298 


93180 


379*9 


92532 


39528 


91856 


4**25 


91152 


43 


IS 


34694 


93789 


3 6 3 2 5 


93169 


37946 


92521 


39555 


91845 


41151 


91140 


42 


19 


347 21 


93779 


36352 


93159 


37973 


92510 


39581 


91833 


4**78 


91128 


41 


20 
21 


34748 


93769 
93759 


36379 
36406 


93148 


37999 
38026 


92499 
92488 


39608 
39635 


91822 


41204 
4*231 


91116 
91104 


40 
39 


34775 


93137 


91810 


22 


34803 


93748 


36434 


93127 


38053 


92477 


39661 


91799 


4*257 


91092 


38 


23 


34830 


93738 


36461 


93116 


38080 


92466 


39688 


9i787i 


41284 


91080 


37 


24 


34^57 


93728 


36488 


93106 


38107 


92455 


39715 


91775! 


4*3*0 


91068 


36 


2o 


34884 


937i8 


36515 


93095 


38134 


92444 


3974 1 


91764 


4*337 


91056 


35 


26 


34912 


93708 


36542 


93084 


38161 


92432 


39768 


9 J 752! 


4*363 


9*o44 


34 


27 


34939^ 


93698 


36569 


93074 


38188 


92421 


39795 


917411 


41390 


91032 


33 


28 


34966 


93688 


36596 


93063 


38215 


92410 


39822 


91729, 


41416 


91020 


32 


29 


34993 


93677 


36623 


93052 


38241 


92399 


39848 


91718I 


4*443 


91008 


31 


30 
31 


35021 
35048 


93667 
93657 


36650 
36677 


93042 


38268 
38295 


92388 
92377 


39875 
39902 


91706 
91694 


4*469 


90996 
90984 


30 

29 


93°3* 


4*496 


32 


35°75 


93647 


36704 


93020 


38322 


92366 


39928 


91683 


41522 


90972 


28 


33 


35102 


93637 


36731 


93010 


38349 


92355 


39955 


91671 


4*549 


90960 


27 


34 


35 I 3° 


93626 


3<>7S8 


92999 


38376 


92343 


39982 


91660 


4*575 


90948 


26 


3D 


35157 


93616 


36785 


92988 


38403 


92332 


40008 


91648 


41602 


90936 


25 


36 


35184 


93606 


36812 


92978 


38430 


92321 


40035 


91636 


41628 


90924 


24 


37 


35211 


93596 


36839 


92967 


38456 


92310 


40062 


91625 


4*655 


90911 


23 


38 


35239 


93585 


36867 


92956 


38483 


92299 


40088 


91613 


41681 


90899 


22 


39 


35266 


93575 


36894 


92945 


38510 


92287 


401 1 5 


91601 


4*707 


90887 


21 


40 
41 


35293 

35320 


93565 
93555 


36921 
36948 


92935 
92924 


38537 
38564 


92276 j 
92265 


1 401 4 1 
40168 


91590 


4*734 
41760 


90875 
90863 


20 

l 
19 


91578 


42 


35347 


93544 


36975 


92913 


38591 


92254| 


40195 


91566 


4*787 


90851 


18 


43 


35375 


93534 


37002 


92902 


38617 


92243 | 


40221 


9*555 


4*813 


90839 


17 


44 


354° 2 


935 2 4 


37029 


92892 


38644 


92231 i 


40248 


9*543 


41840 


90826 


16 


4o 


354 2 9 


935*4 


37°5 6 


92881 


38671 


92220 


40275 


9*53* 


41866 


90814 


15 


46 


35456 


935°3 


37083 


92870 


38698 


92209 


40301 


9*5*9 


41892 


90802 


14 


47 


354H 


93493 


37110 


92859 


38725 


92198 


40328 


91508 


4*9*9 


90790 


13 


48 


355** 


93483 


37137 


92849 


.38752 


92186 j 


40355 


9*496 


4*945 


90778 


12 


49 


35538 


93472 


37164 


92838 


38778 


92175 1 


40381 


9*484 


4*972 


90766 


11 


50 
51 


355 6 5 
3559 2 


93462 
93452 


37*9* 
37218 


92827 
92816 


38805 
38832 


92164 
92152 


40408 
40434 


91472 


11998 
42024 


90753 
90741 


10 
9 


91461I 


52 


35619 


93441 


37245 


92805 


3*859 


92141 


40461 


9*449 


42051 


90729 


8 


53 


35647 


9343 1 


37272 


92794 


38886 


92130 


40488 


9*437 


42077 


907*7 


7 


54 


35674 


93420 


37299 


92784 


38912 


92119 


40514 


9*425 


42104 


90704 


6 


55 


357oi 


93410 


37326 


92773 


38939 


92107 


40541 


9*4*4 


42130 


90692 


5 j 


56 


35728 


93400 


37353 


92762 


38966 


92096 


40567 


91402 


42156 


90680 


4 


57 


35755 


93389 


3738o 


92751 


3^993 


92085 


40594 


9 i 39 oi 


42183 


90668 


3 


58 


35782 


93379 


37407 


92740 


39020 


92073 


40621 


9*378 


42209 


90655 


2 


59 


35810 


93368 


37434 


92729 


39046 


92062 ; 


40647 


9*366 


42235 


90643 


1 1 


60 
/ 


35837 
Cosine. 


93358 
Sine. 


3746i 
Cosine. 


92718 
Sine. 


39073 
Cosine. 


92050' 

Sine. 


40674 


9*355 
Sine. 


42262 
Cosine. 


90631 


1 
/ 


Cosine. 


Sine. 


69° 


68° 


67° 


66° 


65° 



92 



NATURAL SINES AND COSINESS. 


| / 




25° 


26° 


27° 


28° 


29° 


/ 


Sine. 


Cosine. 


Sine. 


Cosine, j 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


42262 


90631 [ 


43 8 37 


89879 


45399 


89101 


46947 


88295 


48481 


87462 


i 


42288 


90618 ' 


43863 


89867 


45425 


89087! 


46973 


88281 


48506 


87448 


59 


2 


42315 


90606 


43889 


89854 


45451 


89074 | 


46999 


88267 


48532 


87434 


58 


3 


42341 


90594 


43916 


89841 


45477 


89061 


47024 


88254 


48557 


87420 


57 


4 


42367 


90582 


43942 


89828 


45503 


89048 


47050 


88240 


48583 


87406 


56 


5 


4 2 394 


90569 


43968 


89816 


45529 


89035 


47076 


88226 


48608 


87391 


55 


6 


42420 


9°557 


43994 


89803 


45554 


89021 


47101 


88213 


48634 


87377 


54 


V 


42446 


9°545 


44020 


89790 


45580 


89008 


47127 


88199I 


4865,9 


87363 


53 


s 


42473 


90532 


44046 


89777 


45606 


88995 


47153 


88185 


48684 


87349 


52 


y 


42499 


90520 


44072 


89764 


45632 


88981 


47178 


88172! 


48710 


87335 


51 


10 


42525 


90507 


44098 


89752 


45658 


88968 


47204 


88158 


48735 


87321 


50 


n 


42552 


90495 


44124 


89739 


45684 


88955 


47229 


88144 48761 


87306 


49 


12 


42578 


90483 


441 5 1 


89726 


45710 


88942 


47255 


88130, 48786 


87292 


48 


13 


42604 


90470 


44*77 


89713 


4573 6 


88928 


47281 


88117 48811 


87278 


47 


! 14 


42631 


90458 


44203 


89700 


45762 


88915 


47306 


88103I 48837 


87264 


46 


lb 


42657 


90446 


44229 


89687 


457 8 7 


88902 


47332 


88089 


48862 


87250 


45 


1 16 


42683 


90433 


44255 


89674 


45813 


88888 


47358 


88075 j 


48888 


87235 


44 


1 iv 


42709 


90421 


44281 


89662 


45 8 39 


88875 


47383 


88062! 


4S913 


87221 


43 


18 


42736 


90408 1 


44307 


89649 


45865 


88862 


47409 


880481148938 


87207 


42 


1 iy 


42762 


90396 


44333 


89636 


45891 


88848 


47434 


880341I48964 


87193 


41 


j 20 


42788 


90383 


44359 


89623 


459 J 7 


88835 


47460 


88020: 48989 


87178 


40 


i 21 


42815 


90371 


443 8 5 


89610 


45942 


88822 


47486 


88006 j 49014 


87164 


39 


1 22 


42841 


90358 


4441 1 


8 9597 


45968 


88808 


475 11 


87993 49040 


87150 


38 


23 


42867 


90346: 


44437 


89584 


45994 


88795 


47537 


87979 ! 49065 


87136 


37 


1 a 


42894 


9°334j 


44464 


89571 


46020 


88782 


47562 


879651 


49090 


87121 


36 


! 2o 


42920 


90321 


44490 


89558 


46046 


88768 


47588 


87951 


49116 


87107 


35 


j 26 


42946 


90309! 


44516 


8 9545 


46072 


88755 


47614 


87937| 


49141 


87093 


34 


; 2V 


42972 


90296 : 


44542 


89532 


46097 


88741 


47639 


87923 49166 


87079 


33 


: 28 


42999 


90284: 


44568 


89519 


46123 


88728 


47665 


87909 ;49 I 9 2 


87064 


32 


29 


43025 


9027I j 


44594 


89506 


46149 


88715 


47690 


87896:49217 


87050 


31 


30 


43°5i 


90259 j 


44620 


89493 


46175 


88701 


47716 


87882J 49242 


87036 


30 


31 


43°77 


90246 i 


44646 


89480 


46201 


88688 


47741 


87868^49268 


87021 


29 


32 


43104 


90233: 


44672 


89467 


46226 


88674 


47767 


87854 149293 


87007 


28 


1 33 


43130 


90221 


44698 


89454 


46252 


88661 


r47793 


87840 49318 


86993 


27 


1 34 


43156 


90208 


44724 


89441 


46278 


88647 


47818 


87826 


49344 


86978 


26 


85 


43182 


9OI96 


4475° 


89428 


46304 


88634 


47844 


87812 


49369 


86964 


25 


j 36 


43209 


90183 


44776 


89415 


46330 


88620 


47869 


87798! 


49394 


86949 


24 


i 8V 


43 2 35 


9OI7I 


44802 


89402 


4 6 355 


88607 


47895 


87784- 


49419 


86935 


23 


38 


43261 


90158 j 


44828 


89389 


46381 


88 593 


47920 


87770 


49445 


86921 


22 


i 8y 


43287 


90146 ' 


44854 


89376 


46407 


88580 


47946 


87756; 4947o 


86906 


21 


1 40 


43313 


90133 | 


44880 


89363 


4 6 433 


88566 


4797 1 


877431 


49495 


86892 


20 


j 41 


4334° 


90120 


44906 


89350 


46458 


88553 


47997 


87729 


49521 


86878 


19 


42 


43366 


90108 


44932 


89337 


46484 


88539 


48022 


87715' 


49546 


86863 


IS 


43 


43392 


9OO95 


44958 


89324 


'46510 


88526 


48048 


87701 j 


4957i 


86849 


17 


44 


434* 8 


90082 


44984 


89311 


46536 


88512 


48073 


87687 


49596 


86834 


16 


! 4o 


43445 


90070' 


45010 


89298 


46561 


88499 


48099 


87673 


49622 


86820 


15 


i 46 


43471 


9OO57; 


45036 


89285 


46587 


88485 


48124 


87659! 


49647 


86805 


14 


1 4/ 


43497 


90045 


45062 


89272 


46613 


88472 


48150 


87645 


49672 


86791 


13 


48 


43523 


9OO3Z 


'45088 


89259 


46639 


88458 


48175 


87631! 


49697 


86777 


12 | 


49 


43549 


90019 


45"4 


89245 


46664 


88445 


48201 


87617 


49723 


86762 


11 1 


50 


43575 


90007 


45140 


89232 


46690 


88431 


48226 


87603 


49748 


86748 


10 ! 


51 


43602 


89994 


1 45166 


89219 


46716 


88417 


48252 


87589! 


49773 


86733 


9 i 


| 52 


43628 


89981 


!45 I 9 2 


89206 


46742 


88404 


48277 


87575 i 


49798 


86719 


8 


53 


43 6 54 


89968 


145218 


89193 


46767 


88390 


48303 


875611 


49824 


86704 


7 


o4 


43680 


89956 


! 45243 


89180 


46793 


88377 


48328 


87546: 


49849 


86690 


6 i 


5o 


43706 


8 9943| 


1 45269 


89167 


46819 


88363 


48354 


875321 


49874 


86675 


5 


56 


43733 


89930 


45295 


8 9*53 


46844 


88349 


48379 


87518J 


49899 


86661 


4 


57 


43759 


89918 


145321 


89140 


46870 


883361 


48405 


87504' 


49924 


86646 


3 


58 


437*5 


89905 


'45347 


89127 


46896 


88322 


48430 


87490: 


4995° 


86632 


2 


o9 


43811 


89892; 


145373 


89114 


46921 


88308 


48456 


87476 : 


49975 


86617 


1 


60 
/ 


43 8 37 
Cosine. 


89879 
Sine. 


45399 
Cosine. 


89101 
Sine. 


46947 
Cosine. 


88295 
Sine. 


48481 
Cosine. 


87462 ; 

Sine, j 


50000 


86603 



/ 


Cosine. 


Sine. 


64° 


63° 


62° 


61° 


60° 



93 



NATURAL SINES AND COSINES. 




/ 



30° 


31° 


32° 


33° 


34° 


/ 

60 




Sine. 


Cosine. 
86603 


Sine. 


Cosine. 


Sine. 
52992 


Cosine. 


Sine. 


Cosine. 
83867 


Sine. 


Cosine. 




50000 


51504 


85717 


84805 


54464 


559*9 


82904 




l 


50025 


86588 


51529 


85702 


53017 


84789 


54488 


83851 


55943 


82887 


59 




2 


50050 


86573 


51554 


85687 


53041 


84774 


545*3 


83835 


55968 


82871 


58 




3 


50076 


86559 


5*579 


85672 


53o66 


84759 


54537 


83819 


55992 


82855 


57 




4 


50101 


86544 


5160J. 


85657 


5309* 


84743 


5456i 


83804 


56016 


82839 


56 




5 


50126 


86530 


51628 


85642 


53**5 


84728 


54586 


83788 


56040 


82822 


55 




6 


5 OI 5* 


86515 


5l 6 53 


85627 


53*4° 


84712 


54610 


83772 


56064 


82806 


54 




V 


50176 


86501 


51678 


85612 


53*64 


84697 


54635 


83756 


56088 


82790 


53 




8 


50201 


86486 


5*703 


85597 


53189 


84681 


54659 


83740 


56112 


82773 


52 




y 


50227 


86471 


51728 


85582 


53214 


84666 


54683 


83724 


56136 


82757 


51 




10 


50252 
50277 


86457 
86442 


5*753 
51778 


85567 


53238 
53263 


84650 
84~6~35 


547o8 
54732 


83708 


56160 
56184 


82741 


50 
49 




85551 


83692 


82724 




12 


50302 


86427 


51803 


85536 


53288 


84619 


54756 


83676 


56208 


82708 


48 




13 


50327 


86413 


51828 


85521 


533*2 


84604 


54781 


83660 


56232 


82692 


47 




14 


50352 


86398 


51852 


85506 


53337 


84588 


54805 


83645 


56256 


82675 


46 




15 


50377 


86384 


51877 


85491 


5336i 


84573 


54829 


83629 


56280 


82659 


45 




16 


50403 


86369 


51902 


85476 


53386 


84557 


54854 


83613 


56305 


82643 


44 




IV 


50428 


86354 


51927 


85461 


534** 


84542 


54878 


83597 


56329 


82626 


43 




18 


5°453 


86340 


51952 


85446 


53435 


84526 


54902 


83581 


56353 


82610 


42 




19 


50478 


86325 


5*977 


8543* 


53460 


845 1 1 


54927 


83565 


56377 


82593 


41 




20 
21 


5 503 
50528 


86310 
86295 


52002 
52026 


85416 


53484 
53509 


84495 
84480 


5495* 
54975 


83549 
83533 


56401 
56425 


82577 
82561 


40 
39 




85401 




22 


50553 


86281 


52051 


85385 


53534 


84464 


54999 


835*7| 


56449 


82544 


38 




23 


50578 


86266 


52076 


85370 


53558 


84448 


55024 


83501 


56473 


82528 


37 




24 


50603 


86251 


52101 


85355 


53583 


84433 


55048 


83485 


56497 


82511 


36 




25 


50628 


86237 


52126 


85340 


53607 


844*7 


55072 


83469 


56521 


82495 


30 




26 


5° 6 54 


86222 


5**5* 


85325 


53632 


84402 


55097 


83453 


56545 


82478 


34 




27 


50679 


86207 


52175 


853*0 


53656 


84386 


55*2* 


83437 


56569 


82462 


33 




28 


50704 


86192 


52200 


85294 


53681 


84370 


55*45 


83421 


56593 


82446 


32 




29 


50729 


86178 


52225 


85279 


53705 


84355 


55*69 


83405 


56617 


82429 


31 




30 
31 


50754 
50779 


86163 
86148 


52250 
52275 


85264 


5373° 
53754 


84339 


55*94 
552i8 


83389 
83373 


56641 
56665 


824J3 

82396 


30 
29 




85249 


84324 




32 


50804 


86133 


52299 


85234 


53779 


84308 


55242 


83356 


56689 


82380 


28 




33 


50829 


86119 


52324 


85218 


53804 


84292 


55266 


83340 


567*3 


82363 


27 




34 


50854 


86104 


5 2 349 


85203 


53828 


84277 


55291 


83324 


56736 


82347 


26 




35 


50879 


86089 


5 2 374 


85188 


53853 


84261 


553*5 


83308 


56760 


82330 


25 




36 


50904 


86074 


5 2 399 


85*73 


53877 


84245 


55339 


83292 


56784 


82314 


24 




37 


50929 


86059 


5 2 4 2 3 


85*57 


53902 


84230 


55363 


83276 


56808 


82297 


23 




38 


50954 


86045 


52448 


85142 


53926 


84214 


55388 


83260 


56832 


82281 


22 




39 


50979 


86030 


5 2 473 


85*27 


5395* 


84198 


554*2 


83244 


56856 


82264 


21 




40 

iTf 


51004 


86015 
86000 


52498 
52522 


85112 
85096 


53975 
54000 


84182 
84167 


55436 
5546o 


83228 
83212 


56880 
56904 


82248 
82231 


20 

19 




51029 




42 


51054 


«59«5 


5 2 547 


85081 


54024 


84151 


55484 


83195 


56928 


82214 


18 




, 43 


51079 


85970 


52572 


85066 


54049 


84*35 


55509 


83*791 


56952 


82198 


17 




44 


5 1 104 


8595b 


52597 


85051 


54073 


84120 


55'533 


83163 


56976 


82181 


16 




46 


51129 


85941 


52621 


85035 


54097 


84104 


55557 


83*47 


57000 


82165 


15 




46 


5"54 


85926 


52646 


85020 


54122 


84088 


5558i 


83131 


57024 


82148 


14 




47 


5**79 


85911 


52671 


85005 


54146 


84072 


55605 


83*15 


57047 


82132 


13 




48 


51204 


85896 


52696 


84989 


54*7* 


84057 


55630 


83098 


1 57071 


82115 


12 




49 


51229 


85881 


52720 


84974 


54*95 


84041 


55654 


83082 


57095 


82098 


11 




50 
l"5T 


5^54 

51279 


85866 
8585I 


52745 


84959 

84943 i 


54220 
54244 


84025 
84009 


55678 
55702 


83066 
83050 


57**9 

57*43 


82082 


10 
9 




52770 


82065 




52 


51304 


85836 


52794 


84928 


[54269 


83994 


55726 


83034 


57167 


82048 


8 




53 


5*3 2 9 


85821 


52819 


84913 


54293 


83978 


5575° 


83017 


57191 


82032 


7 




54 


5*354 


85806 


52844 


84897 


543*7 


83962 


55775 


83001 


57215 


82015 


6 




65 


5*379 


85792 


52869 


84882 


54342 


83946 


55799 


82985 


57238 


81999 


5 




56 


51404 


85777 


52893 


84866 


54366 


83930 


55823 


82969 


57262 


81982 


4 




57 


51429 


85762 


52918 


84851 


5439* 


839*5 


55847 


82953 


57286 


81965 


3 




58 


5H54 


8 5747 


52943 


84836 


544*5 


83899 


5587* 


82936 


573*o 


81949 


2 




59 


5H79 


8 573 2 


52967 


84820 


54440 


83883 


55895 


82920 


57334 


81932 


1 




60 
/ 


51504 
Cosine. 


85717 
Sine. 


52992 


84805 
Sine. 


54464 
Cosine. 


83867 
Sine. 


559*9 

Cosine. 


82904 


57358 


8*9*5 



/ 




Cosine. 


Sine. 


Cosine. 


Sine. 




59° 


58° 


57° 


56° 


55° 





94 



'■■ -"■■"■"■'— .im— ■.-».■— —.—..— — p-^»a 

NATURAL SIEfES AETO COSINES. 


/ 




35° 


36° 


37° 


38° 


39° 


/ 
60 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


5735 8 


81915 


58779 


80902 


60182 


79864 


61566 


78801 


62932 


77715 


1 


57381 


81899 


58802 


80885 


60205 


79846 


61589 


78783 


62955 


77696 


59 


2 


574°5 


81882 


58826 


80867 


60228 


79829 


61612 


78765 


62977 


77678 


58 | 


a 


574 2 9 


81865 


58849 


80850 


60251 


79811 


61635 


78747 


63000 


77660 


57 


4 


57453 


81848 


58873 


80833 


60274 


79793 


61658 


78729 


63022 


77641 


56 


5 


57477 


81832 


58896 


80816 


60298 


79776 


61681 


78711 


63045 


77623 


55 


6 


57501 


81815 


58920 


80799 


60321 


79758 


61704 


78694 


63068 


776o5 


54 


V 


575M 


81798 


58943 


80782 


60344 


7974 1 


61726 


78676 


63090 


77586 


53 


8 


57548 


81782 


58967 


80765 


60367 


79723 


61749 


78658 


63113 


7750* 


52 


9 


57572 


81765 


58990 


80748 


60390 


79706 


61772 


78640 


63^5 


7755° 


51 


10 
11 


5759 6 
57619 


81748 


59014 
59037 


80730 
89713 


60414 
60437 


79688 
79671 


61795 
61818 


78622 


63158 
63180 


7753 1 
775*3 


50 
49 


81731 


78604 


12 


57643 


81714 


59061 


80696 


60460 


79653 


61841 


78586 


63203 


77494 


48 


13 


57667 


81698 


59084 


80679 


60483 


79635 


61864 


78568 


63225 


77476 


47 


14 


57691 


81681 


59108 


80662 


60506 


79618 


61887 


78550 


63248 


77458 


46 


15 


57715 


81664 


59 I 3 I 


80644 


60529 


79600 


61909 


78532 


63271 


77439 


45 


16 


5773 s 


81647 


59*54 


80627 


6o553 


79583 


61932 


78514 


63293 


77421 


44 


17 


57762 


81631 


59178 


80610 


60576 


795 b 5 


61955 


78496 


63316 


77402 


43 


18 


57786 


81614 


59201 


8o593 


60599 


79547 


61978 


78478 


63338 


77384 


42 


19 


57810 


81597 


59225 


80576 


60622 


7953° 


62001 


78460 


63361 


77366 


41 


20 
21 


57833 
57857 


81580 
81563 


59248 
59272 


80558 


60645 
60T68 


79512 
79494 


62024 
62046 


78442 
78424 


63383 
63406 


77347 
77329 


40 
39 


80541 


22 


57881 


81546 


59295 


80524 


60691 


79477 


62069 


78405 


63428 


77310 


38 


23 


57904 


81530 


593 l8 


80507 


60714 


79459 


62092 


78387 


63451 


77292 


37 


24 


57928 


81513 


59342 


80489 | 


60738 


79441 


62115 


78369 


63473 


77273 


36 


2o 


57952 


81496 


593 6 5 


80472 


60761 


79424 


62138 


78351 


63496 


77255 


35 


26 


5797 6 


81479 


59389 


80455 1 


60784 


794o6 


62160 


78333 


63518 


77236 


34 


ti 


57999 


81462 


59412 


80438 


60807 


79388 


62183 


78315 


63540 


77218 


33 


28 


58023 


81445 


59436 


80420 


60830 


79371 


62206 


78297 


63563 


77199 


32 


29 


58047 


81428 


59459 


80403 


60853 


79353 


62229 


78279 


63585 


77181 


31 


30 
31 


58070 


81412 
81395 


59_48_ 2 
595o6 


80386 


60876 
60899 


79335 
79318 


62251 


78261 
78243 : 


63608 
63630 


77162 
77144 


30 

29 j 


58094 


80368 


62274 


32 


58118 


81378 


59529 


8035! 


60922 


79300 


62297 


78225 | 


63653 


77125 


28 


33 


58141 


81361 


59552 


80334 


60945 


79282 


62320 


78206 


63675 


77107 


27 


34 


58165 


81344 


59576 


80316 


60968 


79264 


62342 


78188 


63698 


77088 


26 


35 


58189 


81327 


159599 


80299 


60991 


79247 


62365 


78170 


63720 


77070 


25 


36 


58212 


81310J 


59622 


80282 


61015 


79229 


62388 


78152 


63742 


77051 


24 


37 


58236 


81293! 


59646 


80264 


61038 


79211 


6241 1 


78134 


63765 


77033 


23 1 


38 


58260 


812761 


59669 


80247 


61061 


79193 


62433 


78116 


63787 


77014 


22 | 


39 


58283 


81259 


59 6 93 


80230 


61084 


79176 


62456 


78098 


63810 


76996 


21 ! 


40 
41 


5 8 3°7 
5 8 33° 


81242 
81225 


59716 
59739 


80212 


! 61 I07 

- 

61130 


79158 
79140 


62479 
62502 


78079 
78061 


63832 
63854 


76977 
76959 


20 
19 


80195 


42 


5 8 354 


81208 


59763 


80178 


6II53 


79122 


62524 


78043 


63877 


76940 


18 


43 


58378 


81191 


59786 


80160 


!6n 7 6 


79105 


62547 


78025 


63899 


76921 


17 


44 


58401 


81 174 


59809 


80143 


61199 


79087 


62570 


78007 


63922 


76903 


16 


45 


58425 


8II57 


59832 


80125 


61222 


79069 


62592 


77988 


63944 


76884 


15 i 


46 


58449 


81140 


59856 


80108 


61245 


79051 


62615 


77970 


63966 


76866 


14 


47 


58472 


81123 


59879 


80091 


61268 


79033 


62638 


77952 


63989 


76847 


13 


48 


58496 


81106 


59902 


80073 


61291 


79016 


62660 


77934 


6401 1 


76828 


12 1 


49 


58519 


81089 


59926 


80056 


I61314 


78998 


62683 


77916 


64033 


76810 


11 j 


50 


58543 


81072 


59949 


80038 


61337 


78980 


62706 


77897 


64056 


76791 


10 ! 


51 


58567 


81055 


59972 


80021 


\ 61360 


78962 


62728 


77879 


64078 


76772 


9 


52 


5859 


81038 


59995 


80003 


61383 


78944 


62751 


77861 


64100 


76754 


8 


53 


58614 


81021 


60019 


79986 


61406 


78926 


62774 


77843 


64123 


76735 


7 


54 


58637 


81004 


60042 


79968 


61429 


78908 


62796 


77824 


64145 


76717 


6 


55 


58661 


80987 


60065 


79951 


61451 


78891 


62819 


77806 


64167 


76698 


5 


56 


58684 


80970 


60089 


79934 


61474 


78873 


62842 


77788 


64190 


76679 


4 


57 


58708 


80953 


60112 


79916 


61497 


78855 


62864 


77769 


64212 


76661 


3 


58 


58731 


80936 


60135 


79899 


1 61520 


78837 


62887 


7775 1 


64234 


76642 


2 


69 


58755 


80919 


60158 


79881 


•61543 


78819 


■ 62909 


77733 


64256 


76623 


1 


60 
/ 


58779 


80902 


60182 
Cosine. 


79864 

Sine. 


61566 
Cosine. 


78801 


62932 


77715 
Sine. 


64279 
Cosine. 


76604 
Sine. 




* 


Cosine. 


Sine. 


Sine. 


! Cosine. 


i 54° 


53° 


52° 


51° 


50° 



95 



NATURAL SINES AND COSINES. 


/. 



1 

2 
3 
4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 

27 
28 
29 
30 

31 
32 
33 
34 

35 

36 
37 
38 
39 
40 

41 

I 42 
43 
44 
45 

46 

47 
48 
49 
50 

51 
52 
53 
54 
55 

56 
57 
58 
59 
60 

/ 


40° 


41° 


42° 


43° 


44° 


/ 

60 
59 
58 
57 
56 
55 

54 
53 
52 
51 
50 

49 
48 
47 
46 
45 

44 
43 
42 
41 

40 

39 
38 
37 
36 
35 

34 
33 
32 
31 
30 

29 
28 
27 
26 
25 

24 
23 
22 
21 
20 

19 
18 
17 
16 
15 

14 
13 
12 

11 
10 

9 

8 
7 
6 
5 

4 


a 

2 

1 



/ 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


64279 
64301 

64323 
64346 
64368 
64390 
64412 

64435 
64457 

64479 
64501 

64524 
64546 
64568 
64590 
64612 

64635 
64657 

64679 
64701 
64723 

64746 
64768 
64790 
64812 
64834 
64856 
64878 
64901 
64923 
64945 
64967 
64989 
65011 
65033 
65055 
65077 
65100 
65122 
65144 
65166 

65188 
65210 
65232 

65254 
65276 

65298 
65320 
65342 

65364 
65386 

65408 
65430 
65452 

65474 
65496 

65518 
65540 
65562 
65584 
65606 

Cosine. 


76604 
76586 

76567 
76548 
76530 
76511 

76492 
76473 
76455 
76436 
76417 

76398 
76380 
76361 
76342 
76323 

76304 
76286 
76267 
76248 
76229 

76210 
76192 
76173 
76154 
76135 
76116 
76097 
76078 
76059 
76041 

76022 
76003 

75984 
75965 
75946 

75927 
759o8 
75889 
75870 
75851 
75832 
75813 
75794 
75775 
75756 

75738 

75719 
757oo 
75680 
75661 

75642 
75623 
75604 

75585 
75566 

75547 
75528 

75509 
75490 
75471 

Sine. 


65606 
65628 
65650 
65672 

65694 
65716 

65738 

65759 
65781 
65803 
65825 

65847 
65869 
65891 

65913 
65935 
65956 

65978 
66000 
66022 
66044 

66066 
66088 
66109 
66131 
66153 
66175 
66197 
66218 
66240 
66262 

66284 
66306 
66327 
66349 
66371 

66393 
66414 
66436 
66458 
66480 

66501 
66523 

66545 
66566 
66588 

66610 
66632 
66653 
'66675 
66697 

66718 
66740 
66762 
66783 
66805 

66827 
66848 
66870 
66891 
66913 

Cosine. 


75471 
75452 
75433 
75414 
75395 
75375 
75356 
75337 
753i8 

75299 

75280 

75261 
75241 
75222 
75203 
75184 

75i65 
75H6 
75126 
75107 
75088 

75069 
75050 
75030 
75011 
74992 

74973 
74953 
74934 
74915 
74896 

74876 
74857 
74838 
74818 

74799 
74780 
74760 

74741 
74722 

74703 

74683 
74664 

74644 
74625 
74606 

74586 
74567 
74548 
74528 

74509 
74489 
74470 
74451 
74431 
74412 

74392 
74373 
74353 
74334 
74314 

Sine. 


66913 

66935 
66956 
66978 
66999 
67021 

67043 
67064 
67086 
67107 
67129 

67151 
67172 

67194 
67215 

67237 
67258 
67280 

67301 
67323 
67344 

67366 
67387 
67409 

67430 
67452 

67473 

67495 
67516 

67538 
67559 
67580 
67602 
67623 
67645 
67666 

67688 
67709 
67730 
67752 
67773 

67795 
67816 

67837 
67859 
67880 

67901 
67923 
67944 

67965 
67987 

68008 
68029 
68051 
68072 
68093 

68115 
68136 

68157 
68179 
68200 


743H 
74295 
74276 
74256 
74237 
74217 

74198 
74178 
74159 
74139 
74120 

74100 
74080 
74061 

74041 
74022 

74002 

73983 
73963 

73944 
73924 

73904 
73885 
73865 
73846 
73826 

73806 

73787 
73767 

73747 
73728 

737o8 
73688 
73669 

73649 
73629 

73610 

7359° 
7357° 
73551 
73531 

735" 
73491 
73472 
73452 
73432 

73413 
73393 
73373 
73353 
73333 

733H 
73294 

73274 
73254 
73234 

73215 
73195 
73175 
73155 
73135 

Sine. 


68200 
68221 
68242 
68264 
68285 
68306 

68327 
68349 
68370 
68391 
68412 

68434 
68455 
68476 
68497 
68518 

68539 
68561 

68582 

68603 

68624 

68645 
68666 
68688 
68709 
68730 

68751 
68772 
68793 
68814 
68835 

68857 
68878 
68899 
68920 
68941 

68962 
68983 
69004 
69025 
69046 

69067 

69088 

69109 

69130 

69151 

69172 

69193 
69214 

69235 
69256 

69277 
69298 

69319 
69340 
69361 

69382 

69403 
69424 
69445 
69466 

Cosine. 


73135 
73116 
73096 
73076 
73056 

73036 

73016 
72996 
72976 
72957 
72937 

72917 
72897 

72877 
72857 
72837 
72817 
72797 
72777 
72757 
72737 

72717 
72697 
72677 
72657 
72637 

72617 
72597 
72577 
72557 
72537 

72517 
72497 
72477 
72457 
72437 

72417 
72397 
72377 
72357 
72337 
72317 
72297 
72277 
72257 
72236 

72216 
72196 
72176 
72156 
72136 

72116 
72095 
72075 
72055 
72035 
72015 
71995 
71974 
71954 
71934 

Sine. 


69466 

69487 
69508 

69529 
69549 
69570 

69591 
69612 
69633 

69654 
69675 

69696 

69717 
69737 
69758 
69779 
69800 
69821 
69842 
69862 
69883 

69904 

69925 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 

70112 
70132 

70153 
70174 
70195 

70215 
70236 
70257 
70277 
70298 

70319 

70339 
70360 
70381 
70401 

70422 

70443 
70463 
70484 
70505 

70525 
70546 

70567 
70587 
70608 

70628 
70649 
70670 
70690 
70711 

Cosine. 


71934 
71914 
71894 
71873 
71853 
71833 

71813 

71792 
71772 
71752 

71732 
71711 
71691 
71671 
71650 
71630 

71610 

71590 
71569 

71549 
71529 

71508 
71488 
71468 

7H47 
71427 

71407 
71386 
71366 

71345 
71325 

71305 
71284 
71264 
71243 
71223 

71203 
71182 
71162 
71141 
71121 

71 100 
71080 

71059 
71039 
71019 

70998 
70978 
70957 
70937 
70916 

70896 

70875 
70855 
70834 
70813 

70793 

70772 

70752 

70731 
70711 


Cosine. 


Sine. 


49° 


48° 


47° 


46° 


45° 



96 



TABLE OF CHOKDS. 



« 



A TABLE OF CHORDS. 



M. 


0° 


1° 


2° 


3° 


40 


5° 


6° 


7° 


8° 


M. 







oooo 


.0175 


.0349 


.0524 


.0698 


.0872 


.1047 


.1221 


•1395 


5 


0015 


.0189 


.0364 


.0538 


.0713 


.0887 


.1061 


.1235 


.1410 


5 


10 


0029 


.0204 


.0378 


•o553 


.0727 


.0901 


.1076 


.1250 


.1424 


10 


15 


0044 


.0218 


•°393 


.0567 


.0742 


.0916 


.1090 


.1265 


.1439 


15 


20 


0058 


.0233 


.0407 


.0582 


.0756 


.0931 


.1105 


.1279 


•1453 


20 


25 


0073 


.0247 


.0422 


.0596 


.0771 


.0945 


.1119 


.1294 


.1468 


25 


30 


0087 


.0262 


.0436 


.0611 


.0785 


.0960 


•"34 


.1308 


.1482 


30 


35 


0102 


.0276 


.0451 


.0625 


.0800 


.0974 


.1148 


•i3 2 3 


.1497 


35 


40 


0116 


.0291 


.0465 


.0640 


.0814 


.0989 


.1163 


•1337 


.1511 


40 


45 


0131 


.0305 


.0480 


.0654 


.0829 


.1003 


.1177 


•1352 


.1526 


45 


50 


0145 


.0320 


.0494 


.0669 


.0843 


.1018 


.1192 


.1366 


.1540 


50 


55 


0160 


.0335 


.0509 


.0683 


.0858 


.1032 


.1206 


.1381 


•1555 


55 


60 


0175 


.0349 


.0524 


.0698 


.0872 


.1047 


.1221 


•1395 


.1569 


60 





9° 


10° 


11° 


12° 


13° 


14° 


15° 


16° 


17° 





1569 


.1743 


.1917 


.2091 


.2264 


.2437 


.2611 


.2783 


.2956 


5 


1^4 


.17555 


.1931 


.2105 


.2279 


.2452 


.2625 


.2798 


.2971 


5 


10 


1598 


.1772 


.1946 


.2119 


.2293 


.2466 


.2639 


.2812 


.2985 


10 


15 


1613 


.1787 


.i960 


.2134 


.2307 


.2481 


.2654 


.2827 


.2999 


15 


20 


1627 


.1801 


•1975 


.2148 


.2322 


.2495 


.2668 


.2841 


.3014 


20 


25 


1642 


.1816 


.1989 


.2163 


.2336 


.2510 


.2683 


.2855 


.3028 


25 


30 


.1656 


.1830 


.2004 


.2177 


.2351 


.2524 


.2697 


.2870 


.3042 


30 


35 


1671 


.1845 


.2018 


.2192 


.2365 


.2538 


.2711 


.2884 


.3057 


35 


40 ( 


.1685 


.1859 


.2033 


.2206 


.2380 


•2553 


.2726 


.2899 


.3071 


40 


45 


.j-00 


.1873 


.2047 


.2221 


.2394 


.2567 


.2740 


.2913 


.3086 


45 


50 


1714 


.1888 


.2062 


.2235 


.2409 


.2582 


•2755 


.2927 


.3100 


50 


55 


,729 


.1902 


.2076 


.2250 


.2423 


.2596 


.2769 


.2942 


•3"4 


55 


60 


1743 


.1917 


.2091 


.2264 


• 2 437 


.2611 


.2783 


.2956 


.3129 


60 



98 





TABX.E OF CHORDS. 






M. 

T 


18° 


19° 


20° 

•3473 


21° 


22° 


23° 


24° 


25° 


26° 


M. 






.3129 


.3301 


•3 6 45 


.3816 


•3987 


•4158 


.4329 


•4499 




5 1 


•3*43 


•33i5 


•3487 


•3 6 59 


.3830 


.4002 


.4172 


•4343 


•4513 


5 




10 


•3*57 


.3330 


.3502 


•3 6 73 


•3845 


.4016 


.4187 


•4357 


.4527 


10 




15 j 


.3172 


•3344 


.3516 


.3688 


•3859 


.4030 


.4201 


.4371 


.4542 


15 




20 


.3186 


•3358 


•353° 


.3702 


•3873 


.4044 


.4215 


.4386 


.4556 


20 




25 


.3200 


•3373 


•3545 


.3716 


.3888 


.4059 


.4229 


•4400 


.4570 


25 




30 


•3 2I 5 


•3387 


•3559 


•373° 


.3902 


.4073 


.4244 


.4414 


•4584 


30 




35 


.3229 


.3401 


•3573 


•3745 


.3916 


.4087 


.4258 


.4428 


•4598 


35 




40 


.3244 


.3416 


.3587 


•3759 


•393° 


.4101 


.4272 


.4442 


.4612 


40 




45 


.3258 


•343° 


.3602 


•3773 


•3945 


.4116 


.4286 


.4456 


.4626 


45 




50 


.3272 


•3444 


.3616 


.3788 


•3959 


.4130 


.4300 


.4471 


.4641 


50 




55 


.3287 


•3459 


•3 6 3° 


.3802 


•3973 


.4144 


■43 x 5 


•4485 


.4655 


55 




60 



.3301 


•3473 


•3 6 45 


.3816 


•3987 


.4158 


•43 2 9 


•4499 


.4669 


60 
~0~ 




27° 


28° 


29° 


30° 

•5!76 


31° 


32° 


33° 


34° 


35° 




.4669 


.4838 


.5008 


•5345 


•55i3 


.5680 


•5847 


.6014 




5 


.4683 


•4853 


.5022 


.5190 


•5359 


•55^7 


.5694 


.5861 


.6028 


5 




10 


.4697 


.4867 


.5036 


.5204 


•5373 


•554i 


.5708 


•5875 


.6042 


10 




15 


•47 1 1 


.4881 


.5050 


.5219 


.5387 


•5555 


.5722 


.5889 


.6056 


15 




20 


.4725 


•4895 


.5064 


-52-33 


.5401 


•55 6 9 


•573 6 


•59°3 


.6070 


20 




25 


.4740 


.4909 


.5078 


.5247 


•5415 


.5583 


•575° 


•5917 


.6083 


25 




30 


•4754 


.4923 


.5092 


.5261 


•54^9 


•5597 


.5764 


•593 1 


.6097 


30 




35 


.4768 


•4937 


.5106 


•5 2 75 


•5443 


.5611 


•5778 


•5945 


.6m 


35 




40 


.4782 


•49 5 " 


.5120 


.5289 


•5457 


.5625 


.5792 


•5959 


.6125 


40 




45 


.4796 


.4965 


•5 J 34 


•53°3 


•547i 


•5638 


.5806 


.5972 


.6139 


45 




50 


.4810 


•4979 


.<5i 4 8 


•53 x 7 


•5485 


.5652 


.5820 


.^986 


.6153 


50 




55 


.4824 


•4994 


.5162 


•533 1 


•5499 


.5666 


.5833 


.6000 


.6167 


55 




60 



.4838 


.5008 


.5176 


•5345 


•5513 


.5680 


.5847 


.6014 


.6180 


60 





36° 


37° 


38° 


39° 


40° 


41° 


42° 


43° 


44° 




.6180 


.6346 


•6511 


.6676 


.6840 


.7004 


.7167 


•733° 


.7492 




5 


.6194 


.6360 


.6525 


.6690 


.6854 


.7018 


.7181 


•7344 


.7506 


5 




10 


.6208 


.6374 


■6539 


.6704 


.6868 


.7031 


•7i95 


•7357 


•75 I 9 


10 




15 


.6222 


.6387 


•6553 


.6717 


.6881 


.7045 


.7208 


•737i 


•7533 


15 




20 


.6236 


.6401 


.6566 


.6731 


.6895 


.7059 


.7222 


•7384 


.7546 


20 




25 


.6249 


.6415 


.6580 


.6745 


.6909 


.7072 


•7235 


•7398 


.7560 


25 




30 


.6263 


.6429 


.6594 


.6758 


.6922 


.7086 


.7249 


•74" 


•7573 


30 




35 


.6277 


.6443 


.6608 


.6772 


.6936 


.7099 


.7262 


.7425 


•7586 


35 




40 


.6291 


.6456 


.6621 


.6786 


.6950 


•7"3 


.7276 


.7438 


.7600 


40 




45 


.6305 


.6470 


.6635 


.6799 


.6963 


.7127 


.7289 


.7452 


.7613 


45 




50 


.6319 


.6484 


.6649 


.6813 


.6977 


.7140 


•73°3 


.7465 


.7627 


50 




55 


.6332 


.6498 


.6662 


.6827 


.6991 


•7154 


.7316 


•7479 


.7640 


55 




60 



.6346 


.6511 


.6676 


.6840 


.7004 


.7167 


•733° 


.7492 


.7654 


60 





45° 


46° 


47° 


48° 


49° 


50° 


51° 


52° 


53° 




.7654 


.7815 


•7975 


.8135 


.8294 


.8452 


.8610 


.8767 


.8924 




5 


.7667 


.7828 


.7988 


.8148 


.8307 


.8466 


.8623 


.8780 


•8937 


5 




10 


.768! 


.7841 


.8002 


.8161 


.8320 


•8479 


.8636 


•8794 


.8950 


10 




15 


.7694 


.7855 


.8015 


.8175 


•8334 


.8492 


.8650 


.8807 


.8963 


15 




20 


.7707 


.7868 


.8028 


.8188 


•8347 


.8505 


.8663 


.8820 


.8976 


20 




25 


.7721 


.7882 


.8042 


.8201 


.8360 


.8518 


.8676 


.8833 


.8989 


25 




30 


•7734 


.7895 


•8o<;s 


.8214 


•8373 


•8531 


.8689 


.8846 


.9002 


30 




35 


.7748 


.7908 


.8068 


.8228 


.8386 


.8545 


.8702 


.88s 9 


.9015 


35 




40 


.7761 


.7922 


.8082 


.8241 


.8400 


.8558 


.8715 


.8872 


.9028 


40 




45 


•7774 


•7935 


.8095 


.8254 


.8413 


.8571 


.8728 


.8885 


.9041 


45 




50 


.7788 


•7948 


.8108 


.8267 


.8426 


.8584 


.8741 


.8898 


.9054 


50 




55 


.7801 


.7962 


.8121 


.8281 


•8439 


•8597 


.8754 


.8911 


.9067 


55 




60 


.7815 


•7975 


•8i35 


1 -8294 


.8452 


.8610 


.8767 


| .8924 


.9080 


60 





99 



TABLE OF CHORDS. 


M. 




54° 


55° 


56° 


57° 


58° 


59° 


60° 


61° 


62° 


M. 


.9080 


.9235 


.9389 


•9543 


.9696 


.9848 


1. 0000 


1.0151 


1. 0301 


5 


.9093 


.9248 


.9402 


.955b 


•9709 


.9861 


1. 0013 


1. 0163 


1.0313 


5 


10 


.9106 


.9261 


•94i5 


.9569 


.9722 


.9874 


1.0025 


1. 0176 


1.0326 


10 


15 


.9119 


.9274 


.9428 


.9581 


•9734 


.9886 


1.0038 


1.0188 


i-°33 8 


15 


20 


.9132 


.9287 


.9441 


•9594 


•9747 


.9899 


1.0050 


1. 0201 


1.0351 


20 


25 


•9i45 


.9299 


•9454 


.9607 


.9760 


.9912 


1.0063 


1. 0213 


1.0363 


25 


30 


•9i57 


.9312 


.9466 


.9620 


.9772 


.9924 


1.0075 


1.0226 


i-o375 


30 


35 


.9170 


•93*5 


•9479 


•9 6 33 


•97*5 


•9937 


1.0088 


1.0238 


1.0388 


35 


40 


.9183 


-933 8 


.9492 


.9645 


.9798 


.9950 


I.OIOI 


1. 0251 


1.0400 


40 


45 


.9196 


•935i 


•9505 


. 9 b 5 8 


.9810 


.9962 


1.0113 


1.0263 


1. 0413 


45 


50 


.9209 


•93 6 4 


.9518 


.9671 


.9823 


•9975 


1. 0126 


1.0276 


1.0425 


50 


55 


.9222 


•9377 


•953° 


.9683 


.9836 


.9987 


1.0138 


1.0288 


1.0438 


55 


60 



•9*35 


.9389 


•9543 


.9696 


.9848 
67° 


1. 0000 


1.0151 


1. 0301 


1.0450 


60 



63° 


64° 


65° 


66° 


68° 


69° 


70° 


71° 


1.0450 


1.0598 


1.0746 


1.0893 


1. 1039 


1.1184 


1.1328 


1. 1472 


1.1614 


5 


1.0462 


1.0611 


1.0758 


1.0905 


1.1051 


1.1196 


1. 1340 


1. 1483 


1. 1626 


5 


10 


1.0475 


1.0623 


1. 0771 


1. 0917 


1. 1063 


1. 1208 


1. 1352 


1. 1495 


1. 1638 


10 


15 


1.0487 


1.0635 


1.0783 


1.0929 


1. 1075 


1. 1220 


1-1364 


1. 1507 


1. 1650 


15 


20 


1.0500 


1.0648 


1.0795 


1.0942 


1. 1087 


1. 1232 


1-1376 


1.1519 


1.1661 


20 


25 


1. 0512 


1.0660 


1.0807 


1.0954 


1. 1099 


1. 1 244 


1.1388 


1.1531 


1-1673 


25 


30 


1.0524 


1.0672 


1.0820 


1.0966 


I. XIII 


1. 1256 


1. 1 400 


1. 1543 


1. 1685 


30 


35 


i-o537 


1.0685 


1.0832 


1.0978 


1.1123 


1. 1268 


1.1412 


I-I555 


1. 1697 


35 


40 


1.0549 


1.0697 


1.0844 


1.0990 


1.1136 


1. 1280 


1. 1424 


1. 1567 


1. 1709 


40 


45 


1. 0561 


1.0709 


1.0856 


1. 1002 


1. 1148 


1. 1292 


1. 1436 


1. 1579 


1. 1720 


45 


50 


1.0574 


1. 0721 


1.0868 


1.1014 


1.1160 


1. 1304 


1. 1448 


1. 1590 


1. 1732 


50 


55 


1.0586 


1.0734 


1.0881 


1. 1027 


1.1172 


1.1316 


1. 1460 


1. 1602 


1. 1744 


55 


60 



1.0598 


1.0746 


1.0893 


1-1039 


1.1184 


1. 1328 


1. 1472 


1.1614 


1. 1756 


60 



72° 


73° 


74° 


75° 


76° 


77° 


78° 


79° 


80° 


1. 1756 


1. 1896 


1.2036 


1. 2175 


1. 2313 


1.2450 


1.2586 


1.2722 


1.2856 


5 


1. 1767 


1. 1908 


1.2048 


1. 2187 


1.2325 


1.2462 


1.2598 


1-2733 


1.2867 


5 


10 


1. 1779 


1. 1920 


1.2060 


1. 2198 


1.2336 


1-2473 


1.2609 


1.2744 


1.2878 


10 


15 


1.1791 


1.1931 


1. 2071 


1. 2210 


1.2348 


1.2484 


1.2620 


1.2755 


1.2889 


15 


20 


1. 1803 


1. 1943 


1.2083 


1. 2221 


1-2359 


1.2496 


1.2632 


1.2766 


1.2900 


20 


25 


1.1814 


1. 1955 


1.2094 


1.2233 


1.2370 


1.2507 


1.2643 


1.2778 


1.2911 


25 


30 


1. 1826 


1. 1966 


1. 2106 


1.2244 


1.2382 


1. 2518 


1.2654 


1.2789 


1.2922 


30 


35 


1. 1838 


1. 1978 


1.2117 


1.2256 


1.2393 


1.2530 


1.2665 


1.2800 


1.2934 


35 


40 


1. 1850 


1. 1990 


1. 2129 


1.2267 


1.2405 


1.2541 


1.2677 


1.2811 


1.2945 


40 


45 


1.1861 


1. 2001 


1.2141 


1.2279 


1. 2416 


1.2552 


1.2688 


1.2822 


1.2956 


45 


50 


1. 1873 


1.2013 


1. 2152 


1.2290 


1.2428 


1.2564 


1.2699 


1.2833 


1.2967 


50 


55 


1. 1885 


1.2025 


1. 2164 


1.2302 


1.2439 


1-^575 


1. 2710 


1.2845 


1.2978 


55 


60 



1. 1896 


1.2036 


1.2175 


1.2313 


1.2450 


1.2586 


1.2722 


1.2856 


1.2989 


60 

i 




81° 


82° 


83° 


84° 


85° 


86° 


87° 


88° 


89° 


1.2989 


1.3121 


1.3252 


i-33*3 


1. 3512 


1.3640 


1.3767 


i-3 8 93 


1. 4018 


5 


1.3000 


1. 3132 


1.3263 


1.3393 


1.3523 


1-3651 




377* 


1.3904 


1.4029 


5 


10 


1.3011 


I-3H3 


!-3 2 74 


1.3404 


I -3533 


1. 3661 




3788 


I-39H 


1.4039 


10 


15 


1.3022 


I-3J54 


1.3285 


1. 3415 


J -3544 


1.3672 




3799 


1.3925 


1.4049 


15 


20 


1.3033 


1. 3165 


1.3296 


1.3426 


1-3555 


1.3682 




3809 


J -3935 


1.4060 


20 


25 


1.3044 


1. 3176 


i-33°7 


1-3437 


I-3565 


1.3693 




3820 


1-3945 


1.4070 


25 


30 


i-3°55 


1. 3187 


1.3318 


1-3447 


I-3576 


i-37°4 




3830 


1.3956 


1.4080 


30 


35 


1.3066 


1. 3198 


1.3328 


I-345 8 


i-35 8 7 


I-37H 




3841 


1.3966 


1. 409 1 


35 


40 


1.3077 


1.3209 


1.3339 


1.3469 


1-3597 


1.3725 




3*5i 


1.3977 


1.4101 


40 


45 


1.3088 


1.3220 


1-335° 


1.3480 


1.3608 


J-3735 




3862 


1.3987 


1.4m 


45 


50 


1.3099 


i-3 2 3 J 


i-336i 


1.3490 


1-3619 


I-374 6 




3*72 


1.3997 


1. 4122 


50 


55 


1.3110 


1.3242 


1.3372 


1. 3501 


1.3629 


1-3757 




3883 


1.4008 


1.4132 


55 


60 


1.3121 


1.3252 


i-33 8 3 


1. 3512 


1.3640 


1.3767 


i-3*93 


1. 401 8 


1.4142 


60 



100 






V J 64 8 



LIBRARY OF CONGRESS 



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